FROM-THE-UBRARY-  OF 
WILLIAM  A  HILLEBRAND 


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AN 


ARITHMETIC 


PREPARATORY  SCHOOLS, 
HIGH  SCHOOLS,  AND  ACADEMIES. 


BY 

CHARLES  A.  HOBBS,  A.M., 

Mastbk  of  Mathematics  in  the  Belmont  School,  Belmont,  Mass. 


NEW  YORK: 
A.   LOVELL  &  CO. 

1896. 


hsf 


COPTBIGHT, 

By  Chablbs  A.  Hobbs, 


W'nW^No  ikl^'^-'^ 


p^^ 


PREFACE. 


This  book  is  designed  particularly  for  pupils  in  prepara* 
tory  schools,  and  it  is  likewise  adapted  for  the  use  of  all 
pupils  who  desire  a  thorough  knowledge  of  Arithmetic. 

The  four  fundamental  operations  of  Arithmetic  should 
be  learned  by  a  thorough  drill  in  the  elementary  schools. 
This  cannot  be  too  strongly  insisted  upon,  as  accuracy  and 
readiness  of  work  in  all  parts  of  Arithmetic  are  depend- 
ent on  these  fundamental  operations.  After  such  a  drill 
the  pupil  is  ready  for  rapid  advancement.  Since  this 
fundamental  work  belongs  to  the  elementary  schools,  it 
has  been  omitted  in  this  book,  and  the  space  has  been 
given  to  examples  for  more  advanced  pupils. 

Special  attention  has  been  paid  to  the  selection  of  exam- 
ples, over  a  thousand  of  which  have  been  taken  from 
entrance  papers  given  at  various  universities  and  colleges. 
The  275  miscellaneous  examples  at  the  end  of  the  book 
are  all  taken  from  such  entrance  papers  and  from  entrance 
papers  given  at  the  United  States  Military  and  Naval 
Academies  at  West  Point  and  Annapolis.  An  abundance 
and  variety  of  examples,  sufficient  to  render  the  pupil 
master  of  the  subject,  will  be  found  in  all  parts  of  the 
book. 

In  the  selection  of  illustrative  examples,  great  care  has 
been  taken  to  present  those  which  will  make  clear  to  the 
pupil  all  the  difficulties  he  is  liable  tg  meet.    The  solu- 

995888 


IV  PREFACE. 

tions  are  given  in  full  in  order  that  the  principles  involved 
may  be  clearly  understood  with  but  little  aid  from  the 
teacher. 

The  adaptability  of  a  text-book  to  school  purposes  can 
be  determined  only  by  actual  use  in  the  school-room.  This 
treatise  has  already  stood  this  test,  since  nearly  every  part 
of  it  has  been  used  by  the  author  in  classes,  whose  mem- 
bers have  without  exception  passed  successfully  their  col- 
lege entrance  examinations  in  Arithmetic. 

JSTo  attempt  has  been  made  to  introduce  novel  methods, 
but  in  all  cases  methods  are  given  which  experience  has 
shown  to  be  well  adapted  to  the  needs  of  the  pupil.  In 
the  arrangement  of  subjects,  no  important  departure  has 
been  made.  The  Metric  System,  a  thorough  knowledge  of 
which  is  required  by  all  first-class  universities  and  colleges, 
is  given  directly  after  Compound  Numbers,  and  thereafter 
the  two  systems  are  used  side  by  side,  thus  giving  the  pupil 
a  thoroughly  practical  as  well  as  theoretical  knowledge  of 
the  system. 

It  is  expected  that  the  teacher  will  use  his  judgment 
with  regard  to  omissions.  In  the  endeavor  to  make  the 
book  complete,  certain  subjects  have  been  included  which 
are  not  necessary  to  a  good  knowledge  of  Arithmetic ;  a 
careful  study  of  college  entrance  papers  shows  that  these 
subjects  are  insisted  on  by  some  colleges. 

The  author  desires  to  express  his  thanks  to  the  many 
persons  who  have  aided  him  by  valuable  suggestions,  and 
also  to  the  many  institutions  which  have  responded  so 
promptly  and  often  to  requests  for  entrance  papers. 

CHARLES  A.  HOBBS= 
Belmont,  July  Ist,  1889. 


CONTENTS 


Chapteb  Paob 

I.   Introduction 1 

Casting  out  the  Nines 1 

Casting  out  the  Elevens 3 

Principles  of  Multiplication  and  Division 4 

Separation  of  Terms 4 

II    Decimal  Fractions 7 

Exercises  in  Numeration 8 

Exercises  in  Notation 8 

Addition  of  Decimals 9 

Subtraction  of  Decimals 10 

Multiplication  and  Division  by  10,  100,  1000,  etc 11 

Multiplication  of  Decimals 12 

Contracted  Multiplication 14 

Division  of  Decimals 15 

Contracted  Division 19 

United  States  Money 23 

III.  Factoks 26 

Factoring 28 

Greatest  Common  Divisor 29 

Least  Common  Multiple 32 

Cancellation 36 

IV.  Common  Fractions 38 

Reduction  of  Fractions  to  Lowest  Terms 39 

Reduction   of   Improper   Fractions   to  Whole   or   Mixed 

Numbers 40 

Reduction   of   Whole   or   Mixed   Numbers   to   Improper 

Fractions 41 

Least  Common  Denominator 4? 


VI  CONTENTS. 

Aud'ition  of  Fractions 43 

Subtraction  of  Fractions 44 

Addition  and  Subtraction  of  Fractions  Combined 45 

Multiplication  of  Fractions 46 

Division  of  Fractions 49 

Short  Methods  of  Multiplication  and  Division 50 

Complex  Fractions 52 

To   Find  a  Number  when  a  Fractional   Part  of   it   is 

Known 55 

To  Find  what  Fractional  Part  one  Number  is  of  Another  57 

Reduction  of  Common  Fractions  to  Decimal  Fractions. .  58 

Reduction  of  Decimal  Fractions  to  Common  Fractions. .  59 

Reduction  of  Common  Fractions  to  Circulating  Decimals  59 

Reduction  of  Circulating  Decimals  to  Common  Fractions  61 

■  Greatest  Common  Divisor  of  Fractions 62 

Least  Common  Multiple  of  Fractions 63 

V.   Compound  Numbers 75 

Long  or  Linear  Measure 75 

Surveyors'  Measure 75 

Square  Measure 75 

Cubic  Measure 76 

Wood  Measure 76 

Liquid  Measure  76 

Apotliecjiries'  Fluid  Measure 77 

Dry  Measure 77 

Troy  Weight 77 

Apothecaries'  Weight 77 

Avoirdupois  Weight 78 

Circular  or  Angular  Measure 78 

Measures  of  Time 79 

English  or  Sterling  Money 81 

Miscellaneous  Tables ^ 81 

Reduction  Descending 81 

Reduction  Ascending 83 

Addition  of  Compound  Numbers 86 

Subtraction  of  Compound  Numbers .  , 87 

Difference  between  Dates 89 

Multiplication  of  Compound  Numbers 90 

Division  of  Compound  Numbers ....  91 


CONTENTS.  Vll 

To  Multiply  or  Divide  a  Compound  Number  by  a  Fraction  93 
To  Reduce  a  Fraction  of  one  Denomination  to  Lower 

Denominations 96 

To  Reduce  Lower  Denominations   to  a  Fraction  of  a 

Higher  Denomination 96 

To  Find  what  Fractional  Part  one  Compound  Number  is 

of  Another 98 

To  Reduce  a  Decimal  of  one  Denomination  to  Lower 

Denominations 99 

To   Reduce   Lower  Denominations   to  a  Decimal  of  a 

Higher  Denomination 100 

To  Find  what  Decimal  one  Compound  Number  is  of 

Another 102 

Comparison  of  Weights 102 

Comparison  of  Money 103 

Rectangular  Surfaces 105 

Rectangular  Volumes 109 

VI.   The  Metric  System 116 

Measures  of  Length 116 

Measures  of  Surface 116 

Measures  of  Volume 110 

Measures  of  Capacity 117 

Weight  ■. 117 

Reduction  of  Metric  Numbers 118 

Rectangular  Surfaces  and  Volumes 121 

The  Metric  System  Compared  with  the  Common  System  124 

VII.  Special  Problems 132 

Carpeting  Rooms 132 

Plastering  Rooms 135 

Papering  Rooms 137 

Board  Measure , 139 

Work  Problems 141 

Clock  Problems 144 

Comparison  of  Thermometers 146 

Specific  Gravity 148 

Longitude  and  Time 151 

VIII.  Ratio  and  Proportion 157 

Ratio , 157 


VUl  CONTENTS. 

Simple  Proportion U>9 

Rule  of  Three 161 

Compound  Proportion 164 

Cause  and  Effect 167 

Partitive  Proportion 171 

Simple  Partnership 173 

Compound  Partnership 174 

Averages  or  Alligation 177 

IX.   Percentage 185 

To  Express  a  Rate  Per  Cent  as  a  Common  Fraction  ....   185 
To  Express  a  Common  Fraction  as  a  Rate  Per  Cent  ....    186 

To  Find  any  Per  Cent  of  a  Number 187 

To  Find  the  Base  when  any  Per  Cent  of  it  is  Known  . . .   189 

To  Find  what  Per  Cent  one  Number  is  of  Another 192 

Profit  and  Loss 194 

Commercial  Discount 198 

Commission 201 

Insurance 205 

Taxes 208 

Duties 211 

X.  Interest  and  Discount 216 

Simple  Interest 216 

Exact  Interest 221 

To  Find  the  Rate  Per  Cent 222 

To  Find  the  Time 224 

To  Find  the  Principal ,. 226 

Promissory  Notes , 228 

Partial  Payments  „ 231 

Compound  Interest 236 

Annual  Interest 239 

True  Discount 240 

Bank  Discount 242 

To  Find  the  Face  of  a  Note  to  Yield  a  given  Proceeds. .  245 

Exchange 247 

Domestic  or  Inland  Exchange 249 

Foreign  Exchange 252 

Equation  of  Payments ,    254 

Average  of  Accounts 258 

XI.  Stocks  .,....,...,..,..,,,.,.,,..,.,,.,.,.,,.. 26^ 


.       CONTENTS.  IX 

XII.  Involution  and  Evolution 271 

Involution 271 

Evolution 272 

Square  Root    273 

Cube  Root 278 

Higher  Roots 283 

XIII.  Series 284 

Arithmetical  Progression 284 

Geometrical  Progression 288 

Compound  Interest 292 

Annuities 294 

Annuities  at  Simple  Interest 295 

Annuities  at  Compound  Interest 296 

XIV.  Mensuration 299 

Definitions •     299 

Triangles 301 

Right  Triangles 303 

Quadrilateraii, 306 

Circles 307 

Prisms  and  Cylinders 310 

Pyramids  and  Cones 312 

Spheres  316 

Similar  Surfaces  and  Solids 316 

Miscellaneous  Examples 319 


ARITHMETIC. 

CHAPTER   I. 
INTRODUCTION.  ' 

1.  Before  studying  this  book  the  student  should  be 
perfectly  familiar  with  the  four  fundamental  operations 
of  Arithmetic — Addition,  Subtraction,  Multiplication,  and 
Division.  In  order  to  ensure  accuracy  it  is  always  advisa- 
ble to  test  each  step  of  work. 

In  Addition  add  the  column  downwards  and  then  up- 
wards, and  if  the  results  are  alike,  they  may  be  considered 
correct. 

In  Subtraction  the  best  test  is  to  add  the  subtrahend  and 
the  remainder,  and  if  the  work  is  correct,  the  sum  is  the 
same  as  the  minuend. 

In  Multiplication  and  Division  the  easiest  way  to  test 
the  work  is  simply  to  repeat  each  step ;  however,  if  further 
tests  are  desired,  the  following  can  be  used:  in  Multipli- 
cation divide  the  product  by  the  multiplier,  and  if  the  work 
is  correct,  the  result  is  the  same  as  the  multiplicand ;  in 
Division  multiply  the  quotient  by  the  divisor  and  add  the 
remainder,  if  any,  and  if  the  work  is  correct,  the  result  is 
the  same  as  the  dividend. 

2.  The  fundamental  operations  can  also  be  tested  by  the 
method  known  as  casting  out  the  nines.     The  excess  of 


2  ARITHMETIC. 

nines  is  a  term  used  to  denote  the  remainder  arising  from 
dividing  a  number  by  9. 

10  =  9  +1,  1000  =  9  X  111  +1, 

100  =  9  X  11  + 1,  10000  =  9  X  1111  + 1,  etc. 

We  thus  see  that  a  unit  of  any  order  equals  one  more  than 
9  multiplied  by  some  number.  From  this  it  follows  that 
any  number  of  units  of  any  order  equals  that  number  of 
units  aiddedtjo .'9  multiplied  by  some  number.  For  example, 
.50 c=.9.x. 5  + 5,  600  =  9x66  +  6,  7000=9x777  +  7,  etc. 
"'i'hemfore  ei'ery  ijUmber  consists  of  a  certain  number  of 
nines  increased  by  the  sum  of  its  digits.  For  exa;mple, 
7654  equals  a  number  of  nines  increased  by  7  +  6  +  5+4; 
this  excess  equals  9x2  +  4,  so  that  the  excess  of  nines  is 
4.  This  process  can  be  applied  to  any  number ;  hence  the 
excess  of  nines  in  any  number  equals  the  excess  of  nines  in 
the  sum  of  its  digits. 

Addition. 

82765  ...  1  The  excess  of  nines  in  the  first  number 

4912  ...  7  is  1,  in  the  second  number  7,  in  the  third 

25754  ...  5  number  6,  and  in  the  fourth  number  0. 

6732  ...  0  The  sum  is  13,  and  the  excess  of  nines  is 


4  .  .  .  120163         13  ...  4    4.     The  sum  of  the  numbers  is  120163,  in 
which  the  excess  of  nines  is  4.     The  ex- 
cess is  the  same  in  each  case ;  therefore  the  work  may  be  considered 
correct. 

Subtraction. 

89643 3  The  minuend  equals  the  sum  of  the 

3598  ...     7  subtrahend  and  remainder.     The  excess 

86045  ...    5  o^  nines  in  the  minuend  is  3,     The  excess 

To         Q  of  nines  in  the  subtrahend  is  7,  and  in  the 

remainder  5.     The  excess  in  the  sum  of 

these  two  excesses  is  3,  the  same  as  in  the  minuend ;  therefore  the  work 

may  be  considered  correct. 


INTRODUCTION. 


Multiplication. 

857  ...    2  The  excess  of  nines  in  the  multiplicand  is 

62  ...    8  2;  in  the  multiplier  8.    The  product  is  16,  and 

1714         16  ...  7        *^®  excess  of  nines  is  7.     The  product  of  the 

^14.2  numbers  is  63134,  in  which  the  excess  of  nines 

y  is  7.     The  excess  is  the  same  in  each  case; 

•therefore  the  work  may  be  considered  correct. 


53134 


Division. 

563)87614(155 

87614 

..8 

563 

563... 5 

3131 

155 ...  2 

2815 

349 ...  7 

3164 

5x 

:2  +  7  =  17.. 

..8 

2815 

349 

The  dividend  equals  the  product  of  the  divisor  and  quotient  plus 
the  remainder.  The  excess  of  nines  in  the  dividend  is  8.  The  excess 
of  nines  in  the  divisor  is  6,  in  the  quotient  2,  and  in  the  remainder  7. 
The  product  of  5  and  2  plus  7  equals  17,  in  which  the  excess  is  8,  the 
same  as  in  the  dividend ;  therefore  the  work  may  be  considered  correct. 

3.  The  excess  of  elevens  is  a  term  used  to  denote  the 
remainder  arising  from  dividing  a  number  by  11. 

10  =  11-1,  1000  =  11  X  91  -1, 

100  =  11  X  9  + 1,         10000  =  11  X  909  + 1,  etc. 

We  thus  see  that  a  unit  in  an  odd  place  equals  one  more 
than  11  multiplied  by  some  number,  and  a  unit  in  an  even 
place  equals  one  less  than  11  multiplied  by  some  number. 
From  this  it  follows  that  any  number  of  units  in  an  odd 
place  equals  that  number  of  units  added  to  11  multiplied 
by  some  number,  and  any  number  of  units  in  an  even  place 
equals  that  number  of  units  subtracted  from  11  multiplied 
by  some  number.     Therefore  every  number  consists  of  a 


4  ARITHMETIC. 

certain  number  of  elevens  increased  by  the  sum  of  the 
digits  in  the  odd  places,  decreased  by  the  sum  of  the  digits 
in  the  even  places.  For  example,  45316  equals  a  number 
of  elevens  increased  by  4  -f-  3  +  6,  decreased  by  5  + 1,  and 
the  excess  of  elevens  is  13  —  6,  or  7.  18394  equals  a  num- 
ber of  elevens  increased  by  1  +  3-1-4,  decreased  by  8  -f-  9 ; 
17  cannot  be  subtracted  from  8,  so  one  of  the  elevens  is 
added  to  8,  and  the  excess  of  elevens  is  11  -f-  8  — 17,  or  2. 
This  process  can  be  applied  to  any  number ;  hence  the  excess 
of  elevens  in  any  number  can  be  found  by  subtracting  the  sum 
of  the  digits  in  the  even  places  from  the  sum  of  the  digits  in  the 
odd  places  (increased,  if  necessary,  by  11  or  some  number  of 
IV  s). 

The  processes  of  testing  the  fundamental  operations  by 
casting  out  the  elevens  are  similar  to  those  by  casting  out 
the  nines. 

4.  There  are  a  few  principles  of  Multiplication  and  Divis- 
ion which  should  be  borne  in  mind. 

(1)  Multiplying  either  multiplicand  or  multiplier  by  a 
number  multiplies  the  product  by  the  same  number. 

(2)  Dividing  either  multiplicand  or  multiplier  by  a  num- 
ber divides  the  product  by  the  same  number. 

(3)  Multiplying  the  dividend  or  dividing  the  divisor  by 
a  number  multiplies  the  quotient  by  the  same  number. 

(4)  Dividing  the  dividend  or  multiplying  the  divisor  by 
a  number  divides  the  quotient  by  the  same  number. 

(5)  Multiplying  or  dividing  both  dividend  and  divisor  by 
the  same  number  does  not  change  the  quotient. 

5.  The  student  should  always  take  great  care  in  his  use 
of  signs.  The  signs  +  and  —  always  denote  the  points  of 
separation  in  a  problem,  and  the  parts  between  these  signs 


INTRODUCTION.  5 

are  called  terms.  Each  term  should  be  simplified  by  itself 
before  the  operations  of  addition  and  subtraction  are  per- 
formed. In  the  expression  15  +  12^3—5x2  there  are 
three  terms,  15,  12  -^  3,  and  5x2;  hence  the  value  is 
15_|.4-10  =  9. 

The  parenthesis,  (),  is  used  to  denote  that  the  expression 
contained  therein  should  be  used  as  a  whole.  The  expres- 
sion should  be  simplified  at  first,  and  its  value  substituted. 
For  example,  (18+7-5) -i-(10-6)  +  3x8  =  20-!-4  +  3xB 
=  5  +  24=29. 

Note.  Brackets,   [],  braces,  {},  and  the  vinculum, ,  may  b« 

used  with  the  same  meaning  as  a  parenthesis. 

EXAMPLES. 
Find  the  value  of 

1.  28x4  +  32-i-8-16. 

2.  10  +  6x3-24-*-6  +  12. 

3.  56-J-7  +  12X  3-52-2. 

4.  22-6x3  +  2x5  +  50-h25. 

5.  28x6^14  +  9x8-12  +  42-^-7x3. 

6.  99x8  +  51x10-7x104  +  26. 

7.  99x(8  +  51)xl0-(7xl04  +  26). 

8.  (99  X  8)  +  (51  X  10  -  7)  X  104  +  26. 

9.  (99x8  +  51)xl0-7x(104  +  26). 

10.  (99x8) +  (51x10) -(7x104) +  26. 

11.  99  X  8  +  51  X  (10  -7)  X  104  +  26. 

12.  99  X  (8  +  51)  X  (10  -  7)  X  (104  +  26). 

13.  (105 -i- 21  +  80 -5- 5)  X  (81 +  36 -9). 

14.  (3146  +  279  -  2141)  -  (370  -  263)  +  91  x  3. 


6  ARITHMETIC. 

15.  (2142  -  1729)  X  (3666 -2514) --(354 -5- 6). 

16.  327  X  6  -- 109  +  52  X  5  -  (42  +  8  X  4). 

17.  (46-8)xll+17xl5-f(83x4-327)xl0-39xl4. 

18.  864  -J- 12  -  124  --  (775  ---  25)  +  54  -f-  (61  -  34). 

19.  949  -  13  -  (119  -f-  7  + 1176  -i-  21)  +  3648  ^  32  -  (306 
H51-5  +  672--6). 

20.  From  126+  (16+4)  X 2 take(48-5-2)  +34 x 6--(17-5). 


DECIMAL   FRACTIONS. 


CHAPTER   II. 
DECIMAL   FRACTIONS. 

6.    When  a  number  is  nsed  to  express  whole  units,  it  is 
called  an  integer,  or  integral  number. 

As  in  the  decimal  system  of  notation  a  unit  of  any  order 
has  one  tenth  the  value  of  a  un^t  of  the  next  order  to  the 
left,  we  can  continue  our  notation  toward  the  right  beyond 
units'  place.  To  do  this,  we  place  a  decimal  point  (.)  imme- 
diately after  the  number  in  units'  place,  and  the  next  place 
to  the  right  is  called  tenths;  a  digit  in  this  place  has  one 
tenth  the  value  of  the  same  digit  in  units'  place.  For 
example,  .3  is  read  three  tenths.  The  places  are  continued 
to  the  right  indefinitely,  and  the 
%         i  names  are  given  in  the  table  an- 

I  I      ^  §  nexed. 

The  places  at  the  right  of  the 
decimal  point  are   called    decimal 


5^    S-.D-    2'<» 


9  :5    a 


3  ja    0)3"- 


§  3  «  places,  and  the  numbers  thus  writ- 
n  n  n  0  n  n  0  0  0  ^  ten  are  called  decimal  fractions,  or 
decimals.  They  are  read  like  whole 
numbers,  adding  the  name  of  the  right-hand  place.  For 
example,  .67  is  read  sixty-seven  hundredths.  It  is  to  be 
noticed  that  the  number  consists  of  six  tenths  and  seven 
hundredths,  but  as  six  tenths  is  the  same  as  sixty  hun- 
dredths, the  entire  decimal  is  sixty-seven  hundredths.  .5432 
is  readj^-ye  thousand  four  hundred  thirty-two  ten-thousandths. 
When  whole  numbers  and  decimal  fractions  are  written 
together,  the  word  "and"  should  be  used  to  connect  the 


8 


ABITHMETIC. 


two  parts.    For  example,  362.459  is  read  three  hundred  sixty- 
two  and  four  hundred  Jifty -nine  thousandths. 

Note.  Zeros  may  be  annexed  or  omitted  at  the  right  of  a  decimal 
fraction  without  altering  the  value,  for  either  with  or  without  th«se 
zeros  the  significant  digits  are  in  the  same  decimal  places. 

When  a  decimal  is  written  without  an  integer,  a  zero  may  be  pu*  m 
units'  place,  or  the  place  may  be  left  vacant. 

EXERCISES  IN   NUMERATION. 
Read  the  following  numbers  : 


1.  .6. 

2.  .572. 

3.  .05072. 

4.  .090067. 

5.  .31402. 


6.  .00000041. 

7.  27.8. 

8.  2.78. 

9.  .6008. 

10.  6000.0008. 


11.  41.002. 

12.  750.0081. 

13.  75000.81. 

14.  3.14159. 

15.  28.0097. 


EXERCISES  IN  NOTATION. 
Write  the  following  in  figures : 

1.  Nine  tenths. 

2.  Eighty-two  hundredths. 

3.  Eighty-two  ten-thousandths. 

4.  Three  hundred  sixty-one  thousandths. 

5.  Three  hundred  sixty-one  millionths. 

6.  Nine  thousand  two  hundred  nine  hundred-thousandths. 

7.  Thirty  thousand  six  hundred-thousandths. 

8.  Thirty  thousand  and  six  hundred-thousandths. 

9.  Sixteen  and  three  hundred  forty-one  thousandths. 
10.   One  hundred  fifty-five  millionths. 


DECIMAL  FRACTIONS.  \f 

11.  One  hundred  and  fifty-five  millionths. 

12.  Six  and  five  ten-millionths. 

13.  Sixty-five  ten-millionths. 

14.  Forty-five  and  fifty-eight  hundredths. 
15*.  Three  hundred  twent}^-nine  thousandths. 

16.   Three  hundred  and  twenty-nine  thousandths. 

Addition  of  Decimals. 

7.  Units  of  the  same  order  should  be  written  in  the  same 
column.  This  can  easily  be  accomplished  by  making  the 
decimal  points  fall  under  each  other.  The  process  of 
adding  is  precisely  the  same  as  in  whole  numbers. 

I.   Find  the  sum  of  18.47,  159.363,  70.00451,  and  0.926. 
18.47 
159.0DO  ^j^Q  begin  at  the  right  and  add  as  in  whole  numbers. 

7U.UU4ol         The  decimal  point  comes  under  the  decimal  points  of 
^•^^^  the  problem. 

248.76351 

EXAMPLES. 

1.  Add  together  12.613,  0.00175,  257.8425,  ar-d  0.001345. 

2.  Add  together  17.429,  0.0173,  1156.8,  and  0.0001723. 

3.  Add  together  0.20765,  0.00631,  6758.13247,  and  5.973. 

4.  Add  together  3107.8192,  0.0624, 0.00414,  and  47.2875. 

5.  Add   together  371.87007,   0.00731,   5768.45321,   and 
0.0093. 

6.  Add  together  11.431,  0.00101,  243.342,  400,  and  1.3734. 

7.  Add  together  16.41215,  9.736,  0.00304,  188.24,  and 
29.03069. 


10  ARITHMETIC. 

8.  Add  together  0.61692,  243.734,  901,  68.45213,  and 
8.386. 

9.  Add  ten  thousand  and  one  millionth,  four  hundred- 
thousandths,  ninety-six  hundredths,  and  forty-seven  million 
sixty  thousand  and  eight  billionths. 

10.  Find  the  sum  of  the  following  numbers  :  fifty-seven 
and  three  thousandths,  three  hundred  and  sixty-four  hun- 
dred-thousandths, forty-seven  thousand  and  eight  thousand 
seven  hundred-thousandths,  eighty-seven  millionths,  and 
four  hundred  and  twenty-seven  ten-thousandths. 

Subtraction  of  Decimals. 

8.  The  process  is  the  same  as  in  whole  numbers,  taking 
care  to  have  the  decimal  points  under  each  other. 

I.  From  934.2963  subtract  47.794. 

934.2963  in  the  subtrahend  there  is  no  digit  in  the  place  of  ten- 

47.794         thousandths,  so  we  annex  a  zero  mentally,  and  this  zero 
886  5023      subtracted  from  the  3  of  the  minuend  leaves  3. 

11.  From  35.2  subtract  24.543. 

35.2  In  the  minuend  there  are  no  digits  in  the  places  of 

24.543  hundredths  and  thousandths,  so  we  annex  two  -zeros  men- 

10  657  tally,  and  then  subtract  as  in  whole  numbers. 

EXAMPLES. 

1.  Subtract  284.7654  from  321.07659. 

2.  Subtract  17.2398  from  27.06. 

3.  Subtract  29.9189  from  240.775. 

4.  Subtract  84.736568  from  100.3064231. 

5.  Subtract  49.934  from  500.39. 

6.  Subtract  70.2574  from  365.71. 


DECIMAL  FRACTIONS.  11 

7.  Sul)tract  876.351  from  1000.01. 

8.  Subtract  185.939131  from  186.847. 

9.  Find  the  difference  between  0.0000005  and  0.00005. 

10.  From  ten  take  six  millionths. 

11.  From  two  hundred  and  six  thousandths  take  two 
hundred  six  thousandths. 

12.  What  is  the  value  of  thirty-six  million  minus  thirty- 
six  millionths  ? 

Multiplication  and  Division  by  10,  100,  1000,  etc. 

9.  Since  a  unit  of  any  order  is  ten  times  as  large  as  a 
unit  of  the  next  order  to  the  right,  when  we  move  a  digit 
one  place  to  the  left,  we  multiply  it  by  10.  This  is  the 
same  as  moving  the  decimal  point  one  place  to  the  right. 
For  example,  70  X  10  =  700 ;  85.43  x  10  =  854.3.  If  the  dec- 
imal point  be  moved  two  places  to  the  right,  the  number  is 
multiplied  by  100;  to  multiply  by  any  number  of  lO's, 
the  decimal  point  is  moved  as  many  places  to  the  right  as 
there  are  zeros  in  the  multiplier.  For  example,  0.95  x  100 
=  95 ;  2.5  x  1000  =  2500 ;  0.00043  x  10000  =  4.3. 

For  the  same  reason,  when  we  move  a  digit  one  place  to 
the  right,  we  divide  it  by  10.  This  is  the  same  as  moving 
the  decimal  point  one  place  to  the  left.  For  example, 
26  H-  10  =  2.6 ;  0.49  ^  10  =  0.049.  To  divide  by  any  num- 
ber of  lO's,  the  decimal  point  is  moved  as  many  places  to  the 
left  as  there  are  zeros  in  the  divisor.  For  example,  195-5-100 
=  1.95;  4.53 -- 10000  =  0.000453. 

In  like  manner,  to  multiply  by  0.1,  0.01,  0.001,  etc.,  move 
the  decimal  point  as  many  places  to  the  left  as  there  are 
decimal  places  in  the  multiplier.  To  divide  by  0.1,  0.01, 
0.001,  etc.,  move  the  decimal  point  as  many  places  to  the 
right  as  there  are  decimal  places  in  the  divisor. 


12 


ARITHMETIC. 


Find  the  value  of 

1.  8.7x10. 

2.  0.0069x10. 

3.  95.6x100. 

4.  0.0453  X  100. 
.  5.  4.069  X  1000. 

6.  0.00094x10000. 

7.  9.2-10. 

8.  7.49-^100. 

9.  0.036-100. 

10.  854.3 -r- 1000. 

11.  1.00182 -J- 1000. 

12.  76.541-10000. 


EXAMPLES. 


13.  4.7x0.1. 

14.  8.76x0.01. 

15.  0.0469x0.01. 

16.  0.037x0.001. 

17.  4.62x0.001. 

18.  573.7x0.00001. 

19.  10  -  0.1. 

20.  53.4  -  0.01. 

21.  97.42-0.001. 

22.  0.48-0.001. 

23.  0.1-0.0001. 

24.  7.32-0.00001. 


Multiplication  of  Decimals. 


10.   I.   Multiply  4.92  by  0.3. 


4.92  If  units  of  any  order  be  multiplied  by  an  integer,  the 

0.3  product  consists  of  units  of  the  same  order.  If  4.92  be 
multiplied  by  3,  the  product  is  14.76.  But  the  multiplier 
is  0.3,  a  number  one  tenth  as  large ;  hence  the  product  is 
one  tenth  as  large,  and  the  decimal  point  must  be  moved  one  place  to 
the  left,  giving  1.476  for  the  answer. 


1.476 


II.  Multiply  0.718  by  0.028. 

0.718 
0.028 


5744 
1436 

0.020104 


0.718  multiplied  by  28  would  give  20.104.  But  the 
multiplier  is  only  one  thousandth  of  28 ;  hence  the  deci- 
mal point  must  be  moved  three  places  to  the  left,  giving 
0.020104  for  the  answer. 


DECIMAL  FKACTIONS.  13 

Fvom  tuese  two  examples  we  see  that  we  multiply  as  in 
whole  numbers,  pointing  off  as  many  decimal  places  in  the 
product  as  there  are  in  both  midtiplicand  and  multiplier. 

EXAMPLES. 

1.  Multiply  6.4  by  1.5. 

2.  Multiply  0.64  by  0.15. 

3.  Multiply  0.09  by  0.0016. 

4.  Multiply  0.427  by  345. 

5.  Multiply  76000  by  1.05. 

6.  Multiply  0.076  by  0.0105. 

7.  Multiply  37900000  by  2.005. 

8.  Multiply  0.0379  by  0.2005. 

9.  Multiply  34.27  by  60000. 

10.  Multiply  200.043  by  2.021. 

11.  Multiply  0.785  by  0.0191. 

12.  Multiply  2.708  by  0.007005. 

13.  Multiply  947.36  by  0.00423. 

14.  Multiply  2.708  by  70050000. 

15.  Multiply  8.764  by  40.015. 

16.  Multiply  25.3784  by  12.567. 

17.  Multiply  0.0400268  by  0.260075. 

18.  Multiply  3  hundredths  by  300  thousandths. 

19.  Multiply  five  thousand  and  three  ten-thousandths  by 
five  thousand  three  ten-thousandths. 

20.  Multiply  twelve  thousand  five  hundred  and  six  hun- 
dred seventy-five  millionths  by  four  thousand  sixteen  ten- 
thousandths, 


14  ARITHMETIC. 

21.  Multiply  six  hundred  twenty-five  ten-millionths  by 
three  hundred  and  eight  thousandths. 

Contracted  Multiplication  of  Decimals. 

11.  In  many  examples  in  multiplication  of  decimals,  only 
a  certain  number  of  accurate  decimal  places  are  required  in 
the  product.  All  extra  work  involving  figures  beyond  the 
required  degree  of  accuracy  can  be  avoided  by  the  use  of 
the  following  method : 

Invert  the  order  of  the  figures  of  the  multiplier,  and 
place  them  so  that  the  tenths'  figure  may  be  under  that 
order  of  decimals  to  which  it  is  proposed  to  limit  the  prod- 
uct. Multiply  the  multiplicand  by  each  figure  of  the  mul- 
tiplier, beginning  at  the  figure  immediately  above  it,  and 
taking  in  the  number  carried  from  the  right  hand.  Place 
the  first  figure  of  each  partial  product  in  the  same  column, 
and  add  the  partial  products,  rejecting  the  sum  of  the  right- 
hand  column,  after  carrying  the  nearest  ten. 

I.  Multiply  29.637842  by  85.916,  the  result  to  be  correct 
to  three  decimal  places. 

29.637842  In  order  to  ensure  accuracy  to  tliree  decimal  places, 

61958        the  partial  products  should  be  accurate  to  four  decimal 

23710274        places.     Hence  four  decimal  places  must  be  multiplied 

1481892        ^y  *^®  units'  figure,  and  5  is  placed  under  the  figure 

266740        in  the  fourth  decimal  place ;    this  brings  the  figure 

2964        in  tenths'  place  under  the  order  of  decimals  to  which 

1778        the  product  is  to  be  limited.     As  the  multiplier  is  now 

2546.365  arranged,  each  partial  product  obtained  by  beginning 

to  multiply  at  the  figure  directly  above  contains  four 

decimal  places ;  hence  the  first  figures  of  the  partial  products  are  to  be 

placed  in  the  same  column. 

Begin  at  the  right  to  multiply.  8  X  4  =  32  ;  however,  2  must  be 
carried  from  the  right,  for  8x2  =  16,  which  is  nearer  20  than  10; 
hence  we  set  down  4  and  carry  3  to  the  next  place.  The  process  is 
t^hen  continued  as  in  ordinary  multiplication^ 


DECIMAL  FKACTIONS.  16 

EXAMPLES. 

1.  Multiply  3.7185625  by  2.2134125,  the  result  to  be 
correct  to  two  decimal  places. 

2.  Multiply  8.170663  by  1461.203,  the  result  to  be  cor- 
rect to  three  decimal  places. 

3.  Multiply  78.5126  by  37.8759,  the  result  to  be  correct 

to  four  decimal  places. 

4.  Multiply  375.76843  by  3.14159,  the  result  to  be  cor- 
rect to  four  decimal  places. 

5.  Multiply  13.50629  by  0.36472,  the  result  to  be  cor- 
rect to  five  decimal  places. 

6.  Multiply  5.7203716  by  2.71728,  the  result  to  be  cor- 
rect to  five  decimal  places. 

7.  Multiply  87.896397  by  3.5298875,  the  result  to  be 
correct  to  five  decimal  places. 

8.  Multiply  0.86858896  by  1.0986123,  the  result  to  be 
correct  to  six  decimal  places. 

9.  Multiply  0.69314718  by  0.43429448,  the  result  to  be 
correct  to  seven  decimal  places. 

10.  Multiply  3.1415926  by  itself,  the  result  to  be  correct 
to  seven  decimal  places. 

Division  of  Decimals. 

12.  Since  units  of  any  order  multiplied  by  a  whole  num- 
ber yield  units  of  the  same  order  in  the  product,  units  of 
any  order  divided  by  units  of  the  same  order  yield  a  whole 
number  as  the  quotient ;  in  other  words,  with  equal  decimal 
plac^$  ill  dividerid  and  divisor  the  quotient  is  (i  whole  number. 


16  ARITHMETIC. 

I.  Divide  38.4  by  6. 

Since  the  divisor  is  a  whole  number,  the  quotient 

6)oo.4  jg  j^  y^]iQ\Q  number  as  far  as  the  dividend  is  a  whole 

6.4  number.     Hence  the  decimal  point  in  the  quotient 

is  directly  below  the  decimal  point  of  the  dividend. 

II.  Divide  0.58961  by  0.07. 

Mark  off  by  a  star  as  many  decimal  places  in 
0.07)0.58^961        the  dividend  as  there  are  in  the  divisor.     Then  the 
8.423        quotient  is  a  whole  number  as  far  as  this  star,  and 
the  decimal  point  is  directly  below  the  star. 

III.  Divide  3.12  by  8000. 

8000)  ^003.12  If  the  decimal  point  be  moved  three  places  to 

0.00039        *^^®  ^^^^  ^^  ^°*^^  dividend  and  divisor,  the  quotient 
is  not  changed,  and  the  problem  becomes  .00312 
divided  by  8,  which  equals  .00039.     If,  however,  we  cross  off  the  zeros 
in  the  divisor,  and  place  a  star  three  places  to  the  left  of  the  decimal 
point  in  the  dividend,  the  effect  is  the  same. 

In  Long  Division  the  quotient  can  be  written  above  the  divi- 
dend, and  then  the  decimal  point  is  directly  above  the  star. 

IV.  Divide  82.32  by  2.1. 
39.2 

2.1)82.3^2 

63  Since  there  is  one  decimal  place  in  the  divisor, 

193  the  star  comes  after  the  3,  and  in  the  quotient  the 

189  decimal  point  is  directly  above  the  star. 

"~42 
42 

V.  Divide  0.03969  by  4900. 

0.0000081         Since  there  are  two  zeros  at  the  end  of  the  di- 

4900)^00.03969  visor,  the  star  should  be  placed  two  places  to  the 

392  left  of  the  decimal  point.     Then  divide  by  49,  and 

'    49  the  decimal  point  in  the  quotient  is  directly  above 

49  the  star. 


DECIMAL  FRACTIONS.  17 

VI.  Divide  403920  by  0.00108. 

374000000. 

0.00108)403920.00000^  In  order  to  make  decimal  places  even, 

324  zeros  must  be  annexed  to  the  dividend. 

799  Then  the  division  is  as  before,  and  the 

756  decimal  point  in  the  quotient  is  directly 

432  above  the  star. 
432 

VII.  Divide  0.05  by  4.3  to  five  decimal  places. 
0.01162  -}-  In  this  example  the  divisor  is  not  con- 

4  3'iO  0^50000  tained  an  exact  number  of  times  in  tlie 

43  dividend.    In  all  such  cases   it  is  cus- 

tomary  to  carry  the  division  to  a  certain 

*|^  number  of  decimal  places,  and  then  stop. 

^^  The  answer  can  be  written  0.01 162  +,  the 

270  +  sign  denoting  that  there  is  a  remainder. 

258  If,  however,  the  +  sign  is  omitted,  the 

answer  should  be  written  0.011  Go,  which 
would  be  nearer  correct   than  0.01162, 
because  the  next  figure  would  be  greater 
34  than  5. 

From  these  examples  we  see  that  the  method  for  Division 
of  Decimals  is  as  follows :  If  necessary^  annex  zeros  to  the 
dividend  in  order  to  make  the  decimal  places  equal  in  dividend 
and  divisor.  Then  mark  off  by  a  star  as  many  decimal 
places  in  the  dividerid  as  there  are  in  the  divisor.  Divide  as 
in  whole  numbers,  and  in  the  quotient  place  the  decimal  point 
directly  belotv  or  above  the  star. 

When  the  divisor  is  a  whole  number  ending  in  zeros, 
cross  off  the  zeros,  and  place  the  star  in  the  dividend  as  many 
places  to  the  left  of  the  decimal  point  as  there  are  zeros  in  the 
divisor. 

Note.  In  problems  where  the  divisor  is  not  contained  an  exact 
number  of  times  in  the  dividend,  five  decimal  places  in  the  quotient  is 
ordinarily  far  enough  to  carry  the  division.  If  the  next  digit  is  to  be  less 
than  5,  keep  the  last  digit  as  it  comes  in  the  division ;  if  the  next  diffit  is  to 
he  5,  or  more,  increase  the  last  digit  hy  one. 


120 
86 


18  ARITHMETIC. 

EXAMPLES. 

1.  Divide  769.428  by  200. 

2.  Divide  76.9428  by  0.0002. 

3.  Divide  0.000064  by  0.008. 

4.  Divide  9.00081  by  900. 

5.  Divide  0.000144  by  120000. 

6.  Divide  0.01625  by  0.000025. 

7.  Divide  0.000744  by  0.62. 

8.  Divide  67.56785  by  0.035. 

9.  Divide  0.09  by  0.0016. 

10.  Divide  287.1  by  3300. 

11.  Divide  0.0002548  by  0.0364. 

12.  Divide  0.000647808  by  6.72. 
•     13.  Divide  0.00309824  by  0.376. 

14.  Divide  2926.5  by  0.3902. 

15.  Divide  29.265  by  390.2. 

16.  Divide  1.096641  by  1521. 

17.  Divide  0.0018891  by  3.75. 

18.  Divide  190.914  by  270800. 

19.  Divide  10.85  by  0.0775. 

20.  Divide  3336.894963  by  72530-. 

21.  Divide  0.00091471  by  9.43. 

22.  Divide  189695.4  by  2.708. 

23.  Divide  76.125  by  463000. 

24.  Divide  8.21  by  0.41. 

25.  Divide  0.821  by  410. 


DECIMAL   FRACTIONS.  19 

26.  Divide  0.314  by  1.785. 

27.  Divide  0.10724  by  0.003125. 

28.  Divide  2.838913  by  708.4. 

29.  Divide  0.011825369  by  5.884. 

30.  Divide  695.57270875  by  52.35775. 

31.  Divide  four  millionths  by  four  million. 

32.  Divide  300  thousandths  by  3  hundred-thousandths. 

33.  Divide  sixteen  thousandths  by  forty-five  hundred. 

34.  Divide  eighty-four  and  eighty-four  hundredths  by 
forty-eight  thousandths. 

35.  Divide  fifty  millionths  by  six  hundred  twenty-five 
ten-thousandths. 

36.  Divide  two  thousand  five  hundred  one  and  four 
tenths  by  four  thousand  one  hundred  twenty-five  ten-mil- 
lionths. 

37.  Divide  59285  ten-millionths  by  835  hundred-thou- 
sandths. 

38.  Divide  four  thousand  three  hundred  twenty-two  and 
four  thousand  five  hundred  seventy-three  ten-thousandths 
by  eight  thousand  and  nine  thousandths. 

Contracted  Division  of  Decimals. 

13.  When  examples  in  division  of  decimals  are  required 
to  be  accurate  only  to  a  certain  number  of  decimal  places, 
the  work  can  be  shortened  by  the  use  of  the  following 
method : 

Determine  by  inspection  the  position  of  the  decimal  point 
in  the  quotient,  and  the  number  of  significant  figures  in  the 
quotient  can  at  once  be  determined.     Write  the  divisor  so 


20  ARITHMETIC. 

as  to  contain  two  more  figures  than  the  quotient.  Cut  off 
the  right-hand  figure  of  the  divisor,  and  then  divide;  in 
multiplying  the  divisor  by  the  figure  of  the  quotient,  the 
product  must  be  increased  by  the  number  carried  from  the 
right  hand.  Instead  of  bringing  down  each  time  a  figure 
at  the  right  of  the  remainder,  cut  off  the  right-hand  figure 
of  the  divisor,  and  proceed  as  before. 

I.  Divide  7.97647964  by  3.7876476,  the  result  to  be  cor- 
rect  to  four  decimal  places. 

2.1059  3  is  contained  in  seven  twice ;  hence 

3  787^8^7  976479^64  *^^  quotient  contains  one  integral  place, 

757530  ^"^  *^^  entire  number  of  figures  in  the 

^rv-j-jo  quotient  is  five.     Write  the  first  seven 

S7X76  figures  of  the  divisor,  changing  7  to  8 

on  account  of  the  6  dropped.     Before 

.  gQ .  dividing  cut  off  the  8  in  the  divisor. 

Then  in  multiplying  by  2,  2  must  be 

^'*^  carried  from  the  right  hand,  for  2  X  8 

§^  ^  16,  which  is  nearer  20  than  10.    The 

first  remainder  is   increased  by  1  on 

account  of  the  9  in  the  dividend.     Then  cut  off  the  4  in  the  divisor  aM 

proceed  as  before. 

Note.  If  necessary,  zeros  can  be  annexed  to  the  divisor  to  make  tho 
required  number  of  figures  ;  zeros  occurring  before  the  first  significant 
figure  are  not  to  be  counted. 

EXAMPLES. 

1.  Divide  3698.779375  by  375.625,  the  result  to  be  cor. 
rect  to  three  decimal  places. 

2.  Divide  0.046  by  0.00762089,  the  result  to  be  correct 
to  four  decimal  places. 

3.  Divide  0.32165  by  0.003516,  the  result  to  be  correct 
to  four  decimal  places. 

4.  Divide  0.765439  by  359.21,  the  result  to  be  correct  to 
five  decimal  places. 


DECIMAL  FRACTIONS.  21 

5.  Divide  0.22165  by  0.0035216,  the  result  to  be  correct 
to  five  decimal  places. 

6.  Divide  6.38572164  by  0.0752681,  the  result  to  be  cor- 
rect to  five  decimal  places. 

7.  Divide  29.48495554  by  378.6725,  the  result  to  be  cor- 
rect to  six  decimal  places. 

8.  Divide  100.016  by  3.056,  the  result  to  be  correct  to 

six  decimal  places. 

9.  Divide  0.765439  by  359.21,  the  result  to  be  correct  to 
seven  decimal  places. 

10.   Divide  2.71828128  by  3.1415926,  the  result  to  be  cor- 
rect to  seven  decimal  places. 

MISCELLANEOUS    EXAMPLES. 

1.  Simplify  8.763  -  4.12  +  78.326  -t- 1.1126  -  68.0816. 

2.  Simplify  198.63  +  21.3711  -  100.416  -  45.79  -f  8.3. 

3.  Simplify 

8.72  X  5.4  + 196  x  0.004  -  6.25  x  4.8  -  0.06  x  21.7. 

4.  Simplify 

3.71  X  8  +  2.64  -- 160  -f  7.55  x  0.07  +  0.071  x  25. 

5.  Simplify 

84  X  1.13-  (66-1.2 X 2.4)  -f-100 x  (4 x 0.018+0.189). 

6.  Simplify 

94.5  -  250  + 16^-.  (4.5  --0.225)  +87.25^  (1.6-  0.35). 

7.  Simplify  (15  -  10  x  0.3)  x  6.192  -  (7  x  5.4-35.048). 

8.  The  difference  between  two  numbers  is  94.32^  and 
the  smaller  is  147.631 ;  find  the  larger. 

9.  Divide  the  sum  of  four  thousandths  and  four  mil- 
lionths  by  their  difference. 


22  ARITHMETIC. 

to.  Divide  876.196  by  2.12.  If  the  decimal  point  were 
moved  in  the  dividend  two  places  to  the  left,  and  in  the 
divisor  one  place  to  the  right,  how  many  times  greater  or 
less  would  the  quotient  be  ? 

11.  Miiltiply  forty-eight  ten-thousandths  by  two  and  one 
thousandth,  and  divide  the  result  by  one  million. 

12.  Divide  375  by  0.75  and  0.75  by  375,  and  find  the  sum 
and  difference  of  the  quotients. 

13.  The  product  of  three  numbers  is  5.76 ;  one  of  them  is 
0.024,  and  another  is  0.06 ;  find  the  third. 

14.  The  product  of  three  numbers  equals  70.04597 ;  two 
fi  them  equal  3.91  and  3.0005  respectively ;  find  the  third. 

15.  What  number  divided  by  28.15  will  give  1.216  as  the 
quotient  and  1.5195  as  the  remainder  ? 

16.  The  dividend  is  7423.973,  the  quotient  is  12.13,  and 
the  remainder  is  0.413 ;  what  is  the  divisor  ? 

17.  Find  the  number  of  rods  of  fence  necessary  to  enclose 
a  field,  the  sides  of  which  are  respectively  42.78  rods,  51.3 
rods,  27  rods,  and  37.22  rods. 

18.  Two  men  walk  respectively  26.7  miles  and  22.94 
miles  per  day  ;  how  much  further  does  the  first  walk  than 
the  second  ? 

19.  If  the  year  is  considered  as  365.25  days  instead  of 
365.242264  days,  how  great  will  be  the  error  in  1880  years  ? 

20.  From  a  tank  containing  1200  gallons,  22.75  barrels  of 
31.5  gallons  each  were  pumped  out;  how  many  gallons 
remained  ? 

21.  How  many  barrels,  each  containing  44.5  gallons,  can 
be  filled  from  16554  gallons  of  oil  ? 

22.  4  cords  of  wood  are  worth  as  much  as  13.4  bushels  of 
rye ;  how  much  rye  can  be  obtained  for  15  cords  of  wood  ? 


DECIMAL  FRACTIONS.  23 

23.  A  merchant  bought  972  bushels  of  wheat ;  how  many 
bins,  each  containing  16.25  bushels,  will  be  filled,  and  how 
much  remains  ? 

24.  Two  wheels  of  a  carriage  are  respectively  13.5  feet 
and  11.75  feet  in  circumference ;  how  much  oftener  does  one 
turn  than  the  other  in  going  4000  feet  ? 

United  States  Money. 

14.  The  money  in  use  in  the  United  States  is  expressed 
in  a  decimal  system,  of  which  the  unit  is  a  dollar.  The 
symbol  for  dollars  ($)  is  placed  before  the  number  used  to 
represent  them.  One  dollar  equals  100  cents  (cts,),  and 
cents  are  written  as  a  decimal  fraction  of  a  dollar.  For 
example,  $6.43  means  six  dollars  and  forty -three  cents. 
There  are  three  other  denominations  sometimes  mentioned, 
which  are  not  in  common  use,  —  ten  dollars  equal  one  eagle, 
ten  cents  equal  one  dime,  and  one  tenth  of  a  cent  is  a  mill. 
All  operations  in  United  States  Money  (sometimes  called 
Federal  Money)  are  performed  as  in  decimal  fractions. 

Note  1.  In  general  when  the  final  result  in  a  problem  contains  mills, 
if  less  than  5  they  are  rejected,  if  5  or  more  they  are  called  another 
cent.  If,  however,  the  final  result  is  the  value  of  one  article  where  a 
number  are  considered,  any  fraction  of  a  cent  should  be  retained. 

Note  2.  In  business  transactions  C  is  often  used  for  hundred,  and  M 
for  thousand,  when  the  price  is  by  the  hundred  or  by  the  thousand. 

EXAMPLES. 

1.  A  farmer  sold  24  cows  for  $32.25  apiece ;  how  much 
did  he  receive  ? 

2.  A  drover  sold  42  hogs  for  $246.75 ;  how  much  apiece 
did  he  receive  ? 


24  ARITHMETIC. 

3.  A  merchant  paid  $52  for  a  lot  of  cloth  at  8  cts.  a 
yard ;  how  many  yards  did  he  buy  ? 

4.  A  merchant's  receipts  for  a  week  were  as  follows : 
Monday,  $102.79;  Tuesday,  f72.73;  Wednesday,  $150.65; 
Thursday,  $127.70 ;  Friday,  $205 ;  Saturday,  $278.92.  Find 
the  amount  of  his  receipts  for  the  entire  week. 

5.  A  clerk  has  a  yearly  salary  of  $1000 ;  he  pays  $312 
for  board,  $157.50  for  clothing,  and  $372.25  for  all  other 
expenses.     How  much  does  he  save  in  a  year  ? 

6.  Bought  three  loads  of  wood,  containing  respectively 
2.15  cords,  1.98  cords,  and  1.625  cords ;  find  the  cost  at  $3.15 
a  cord. 

7.  Find  a  man's  daily  wages  when  he  was  paid  $29.70 
for  22  days'  work. 

8.  At  $12,375  a  ton,  how  many  tons  of  hay  can  be 
bought  for  $2326.50  ? 

9.  Bought  3  pounds  of  tea  at  72  cts.  a  pound,  8  pounds 
of  coffee  at  28  cts.  a  pound,  and  15  pounds  of  rice  at  6  cts. 
a  pound ;  find  the  amount  of  the  bill. 

10.  If  $31.75  be  paid  for  5  barrels  of  flour,  what  would 
28  barrels  cost  at  the  same  rate  ? 

11.  If  0.62  of  a  ton  of  hay  be  worth  $11.47,  find  the  value 
of  8.75  tons. 

12.  A  farmer  sold  in  one  month  62  pounds  of.  butter  at 
28  cts.  a  pound,  45  dozen  eggs  at  18  cts.  a  dozen,  and  27 
chickens  at  55  cts.  apiece ;  find  the  amount  of  his  receipts. 

13.  A  lady  bought  12  yards  of  crash  at  14  cts.  a  yard, 
and  8  yards  of  cotton  cloth  at  18  cts.  a  yard,  and  gave  a  $5 
bill  in  payment ;  how  much  change  should  she  receive  ? 

14.  If  the  price  of  gas  be  $1.75  per  M,  find  the  amount 
of  a  man's  bill  when  12240  cubic  feet  have  been  consumed. 


DECIMAL   FKACTIONS.  25 

15.  Sold  7250  cigars  at  $4.20  per  C;  find  the  amount 
received. 

16.  Paid  $10.44  for  1440  bricks  j  what  was  the  price 
per  M  ? 

17.  A  pedler  sells  beets,  six  in  a  bunch,  at  10  cts.  a  bunch, 
and  gains  1  ct.  on  each  bunch ;  find  the  cost  per  C. 

18.  How  many  tons  of  coal  at  $4.75  a  ton  must  be  given 
in  exchange  for  19  barrels  of  flour  at  $6.25  a  barrel  ? 

19.  How  many  dozen  eggs  at  18  cts.  a  dozen  must  be 
given  in  exchange  for  28  pounds  of  sugar  at  11  cts.  a 
pound  and  8  pounds  of  coffee  at  29  cts.  a  pound  ? 

20.  A  merchant  bought  a  load  of  grain  for  $50,  and  by 
retailing  it  at  $1.20  a  bushel,  he  gained  $22;  how  many 
bushels  were  there  in  the  load  ? 

21.  To  send  a  telegram  from  New  York  to  Boston  costs 
25  cts.  for  10  words  and  2  cts.  for  each  additional  word ; 
find  the  cost  of  a  telegram  containing  28  words. 

22.  A  grocer  bought  16  barrels  of  sugar,  each  containing 
232  pounds,  for  $335,  and  sold  it  at  10  cts.  a  pound ;  how 
much  was  his  gain  ? 

23.  Bought  a  roll  of  carpet,  containing  82  yards,  for  $45, 
and  sold  it  for  75  cts.  a  yard ;  find  the  amount  of  profit. 

24.  Bought  a  horse  for  $125,  a  carriage  for  $140,  and  a 
harness  for  $18 ;  kept  them  a  month  at  an  expense  of  $17.25, 
and  then  sold  the  team  for  $300.  Did  I  gain  or  lose,  and 
how  much  ? 

25.  A  merchant  bought  150  barrels  of  apples  for  $300 ;  he 
sold  seven  tenths  of  them  at  $2.25  a  barrel,  and  the  re- 
mainder at  $1,875  a  barrel.  Did  he  gain  or  lose,  and  how 
much? 


26  ARITHMETIC, 


CHAPTER  III. 
FACTORS. 

15.  A  number  which  can  be  contained  in  another  without 
a  remainder  is  called  a  divisor  or  factor  of  that  number. 
When  a  number  has  no  factor  except  itself  and  one,  it  is 
called  a  prime  number ;  when  it  has  other  factors  besides 
itself  and  one,  it  is  called  a  composite  number.  When  two 
numbers  have  no  common  factor  except  one,  they  are  said 
to  be  prime  to  each  other. 

Numbers  of  which  2  is  a  factor  are  called  even  numbers ; 
all  others  are  odd  numbers. 

When  a  number  is  applied  to  some  particular  object  or 
objects,  it  is  called  a  concrete  number;  when  not  applied  to 
any  object,  it  is  called  an  abstract  number.  For  example, 
4  and  7  are  abstract  numbers,  but  4  boys  and  7  books  are 
concrete  numbers. 

An  exponent,  or  index,  is  a  small  figure  placed  at  the 
upper  right-hand  corner  of  a  number  to  show  how  many 
times  it  is  used  as  a  factor.     For  example,  5* =5x5x5x5. 

16.  For  determining  at  sight  whether  certain  numbers 
are  contained  in  a  given  number,  the  following  tests  can  be 
used: 

(1)  A  number  is  divisible  by  2  if  its  right-hand  figure  is 
zero  or  an  even  digit. 

(2)  A  number  is  divisible  by  3  if  the  sum  of  its  digits  is 
divisible  by  3.  For  example,  in  741,  7  -f  4  -f- 1  =  12,  which 
is  divisible  by  3  j  hence  741  is  divisible  by  3. 


FACTORS.  27 

(3)  A  number  is  divisible  by  4  if  the  two  right-hand 
figures  are  zeros,  or  if  the  number  expressed  by  them  is 
divisible  by  4. 

(4)  A  number  is  divisible  by  5  if  its  right-hand  figure  is 
0  or  5. 

(6)  A  number  is  divisible  by  6  if  it  is  an  even  number, 
and  at  the  same  time  is  divisible  by  3. 

(6)  A  number  is  divisible  by  8  if  its  three  right-hand 
figures  are  zeros,  or  if  the  number  expressed  by  them  is 
divisible  by  8. 

(7)  A  number  is  divisible  by  9  if  the  sum  of  its  digits  is 
divisible  by  9. 

(8)  A  number  is  divisible  by  10  if  its  right-hand  figure  is  0. 

(9)  A  number  is  divisible  by  11  if  the  sums  of  the  alter- 
nate digits  are  the  same,  or  if  the  difference  between  these 
sums  can  be  divided  by  11 .  For  example,  in  7458,  7  +  5 
=  4  +  8 ;  hence  7458  is  divisible  by  11.  In  19382,  l-f-3-f-2 
=  6,  9  4-  8  =  17,  and  the  difference  between  6  and  17  is  11 ; 
hence  19382  is  divisible  by  11. 

17.  To  find  whether  a  number  is  prime  or  composite, 
divide  by  the  prime  nup\bers  in  succession  until  one  of 
them  is  contained  in  the  number,  or  else  the  quotient  is  less 
than  the  divisor.  In  the  former  case  the  number  is  compos- 
ite ;  in  the  latter  case  t>>iB  number  is  prime.  For  example, 
take  491.  Dividing  in  succession  by  2,  3,  5,  7,  11,  13,  17, 
and  19,  none  of  them  ar?  contained  in  491,  and  in  every  case 
the  quotient  is  greater  than  the  divisor.  Divide  by  23,  and 
the  quotient  is  21  with  a  remainder ;  hence  491  is  a  prime 
number.  If  the  number  were  composite,  both  divisor  and 
quotient  would  be  factors.  We  have  seen  that  491  has  no 
factor  less  than  23,  and  as  the  divisors  grow  greater,  the 
quotio^:>ts  grow  less  and  will  be  less  than  23.     Then  since 


28  ARITHMETIC. 

there  is  no  factor  less  than  23,  in  no  case  will  the  quotient 
be  a  whole  number,  and  there  are  no  prime  factors  for  the 
number. 

Factoring. 

18.  The  prime  numbers  which  multiplied  together  pro- 
duce a  given  composite  number  are  called  the  prime  factors 
of  that  number. 

I.   Find  the  prime  factors  of  182. 

2^  182  Divide  by  2,  the  least  number  that  is  a 

ijTKT  factor  of  182.    Then  divide  the  quotient 

— r^  by  7,  and  we  find  that  the  three  prime 

factors   of  182   are  2,   7,  and   13.    For 

182  =  2  X  7  X  13.  convenience  of  work  it  is  best  always  to 

divide  by  the  least  possible  factor. 


II.     Find  the  prime  factors  of  3465. 

3)3465 
3)1155 
5)385  When  the   same  factor  occurs  more 

jyfj  than  once,  it  is  best  to  write  that  factor 

~7T  with  an  exponent. 

3465  =  32  X  5  X  7  X  11. 

EXAMPLES. 
Find  the  prime  factors  of  the  following  numbers : 

1.  176.                        7.    792.  13.  4800. 

2.  210.                        8.    1221.  14.  6902. 

3.  360.                       9.   1836.  15.  8364. 

4.  384.                     10.   1872.  16.  10917. 

5.  432.                      11.   2310.  17.  37125. 

6.  48C                     12.   2346.  18.  179487. 


FACTORS.  29 

19.  Which  of  the  numbers  5,  9,  13,  18,  21,  and  25  are 
prime  numbers  ?  Which  of  them  are  prime  to  the  num- 
ber 10? 

20.  Select  the  prime  numbers  between  50  and  100. 

21.  Make  a  list  of  all  the  prime  numbers  below  40,  and 
use  it  to  prove  that  541  is  prime. 

22.  Which  of  the  numbers  293,  371,  385,  440,  524,  617, 
and  713  are  prime  ? 

23.  Of  what  number  are  2,  3,  5,  7,  11,  and  13  the  prime 
factors  ? 

24.  How  many  of  the  different  divisors  of  150  are  prime, 
and  how  many  are  composite  ? 

25.  Find  all  the  prime  factors  common  to  1001  and  616. 

Greatest  Common  Divisor. 

19.  A  common  divisor  of  two  or  more  numbers  is  a  num- 
ber that  will  be  contained  exactly  in  each  of  them. 

The  greatest  common  divisor  of  two  or  more  numbers  is 
the  greatest  number  that  will  be  contained  exactly  in  each 
of  them.  For  example,  3  and  4  are  common  divisors  of  24 
and  36,  but  12  is  the  greatest  common  divisor. 

For  convenience  G.C.D.  is  used  to  represent  the  greatest 
common  divisor. 

Greatest  common  measure  (G.C.M.)  and  highest  common 
factor  (H.C.F.)  are  expressions  which  have  the  same  mean- 
ing as  greatest  common  divisor. 

I.   Find  the  G.C.D.  of  56,  84,  and  140. 

The  G.C.D.  is  the  product  of  all  the 

56  =  2   X  7.  common  prime  factors.     2  and  7  are  the 

84  =  2   X  o  X  7.         common  factors,  but  2  occurs  twice  in  each 

140  =  2   X  O  X  7.         number,  and  hence  will  occur  twice  in  the 

CCD  :=2*X7  =  28        G.C.D.     The  G.C.D.  is  thus  seen   to  bp 

'      ^2^7,  which  equals  2§. 


2)56 

84  140 

2)28 

42     70 

7)14 

21     35 

30  ARITHMETIC. 

To  find  the  G.C.D.  of  two  or  more  numbers,  resolve  th^ 
numbers  into  their  prime  factors,  and  find  the  product  of  the 
common  factors,  taking  each  factor  the  least  number  of  times 
it  occurs  in  any  number. 

Th.Q  following  arrangement  of  work  may  be  used : 

Arrange  the    numbers    in  a  line,  and 
divide  by  all  the  prime  numbers  that  will 
be   contained  in  all   the  numbers.      The 
"^        o        K  divisors  are   the  common  prime  factors, 

and  the  product  of    the   divisors  is  the 
G.C.D.  =  22  X  7  =  28.     G.C.D. 

EXAMPLES. 
Find  the  G.C.D.  of 

1.  48  and  120.  12.  105,  231,  and  1001. 

2.  84,  126,  and  140.  13.  156,  234,  and  260. 

3.  48,  26,  72,  and  24.  14.  189,  243,  and  297. 

4.  6,  8,  20,  36,  and  48.  15.  240,  560,  and  616. 

5.  45, 75, 90, 135, 150,  and  180.     16.  252,  315,  420,  and  504. 

6.  66,  78,  102,  and  114.  17.  256,  480,  and  1296. 

7.  66,  308,  and  506.  18.  432  and  1872. 

8.  119  and  231.  19.  936  and  2925. 

9.  168,  192,  and  216.  20.  720,  336,  and  1736. 

10.  120,  228,  and  720.  21.   927,  342,  and  861. 

11.  144  and  780.  22.   252, 588, 924,  and  1092. 

23.  4815,  4905,  and  5085. 

24.  1209,  1885,  2457,  2691,  and  2717. 

25.  Find  all  the  common  divisors  of  225,  2025,  and  8100. 

26.  What  is  the  length  of  the  longest  boards  that  will 
exactly  fit  three  floors  42;  63;  and  105  feet  long  respectively  ? 


FACTORS.  31 

27.  A  Mian  has  three  farms  of  56,  72,  and  88  acres  respec- 
tively, and  wishes  to  fence  them  into  the  largest  possible 
fields,  having  each  the  same  number  of  acres.  How  many 
acres  could  he  put  in  each  ? 

28.  How  many  gallons  are  there  in  the  largest  vessel  that 
will  exactly  measure  the  contents  of  three  hogsheads,  con- 
taining respectively  143,  104,  and  156  gallons  ? 

29.  rind  the  length  of  the  longest  pole  that  will  exactly 
measure  the  sides  of  a  field,  which  are  respectively  72,  126, 
162,  and  90  feet. 

30.  Three  military  companies  consisting  respectively  of 
36,  42,  and  54  men  are  divided  into  squads,  each  containing 
the  same  number;  find  the  largest  number  of  men  each 
squad  may  contain,  and  the  number  of  squads  in  each 
company. 

20.  When  the  numbers  cannot  be  factored  easily,  a  differ- 
ent method  is  employed  for  finding  the  G.C.D. 

I.   Find  the  G.C.D.  of  161  and  368. 

161)368(2  Divide  368  by  161.    If  161  were  contained 

322  exactly,   it  would  be   the   G.C.D.     Howerer, 

46'\16ir3  there  is  a  remainder  46.     Divide  161  by  46, 

•j^gg  and  there  is  a  remainder  23.     Divide  46  by  23, 

'~oQ\Aa/o       *"^  there  is  no  remainder.    23,  the  last  divisor, 

^^^f^^       is  the  G.C.D. 

—  Since  23  is  a  divisor  of  46,  it  is  a  divisor  of 

138,  which  equals  46x3,  and  is  then  a  divisor  of  161,  which  equals 
138  +  23.  Since  23  is  a  divisor  of  161,  it  is  a  divisor  of  322,  which 
equals  161  X  2,  and  is  then  a  divisor  of  -368,  which  equals  322  +  23x2. 
Hence  23  is  a  common  divisor  of  161  and  368. 

Furthermore,  161  and  368  are  each  a  certain  number  of  times  the 
G.C.D.  161  X  2  =  322,  which  is  a  certain  number  of  times  the  G.C.D. 
368  —  322  =:  46,  which  must  be  a  certain  number  of  times  the  G.C.D., 
because  if  a  number  is  a  divisor  of  two  other  numbers,  it  is  a  divisor  of 
their  difference.    46x3=138,  which  js  Vk  certaip  nwmber  pf  tiwes  tb« 


32  ARITHMETIC. 

G.C.D.  Then  161  —  138,  which  equals  23,  is  a  certain  number  of  times 
the  G.C.D.,  and  the  G.C.D.  cannot  be  greater  than  23.  But  it  has  been 
proved  that  23  is  a  common  divisor ;  hence  it  is  the  G.C.D, 

The  method  may  be  stated  as  follows :  Divide  the  greater 
7iumher  by  the  less,  and  the  divisor  by  the  remainder,  and  so  on 
till  there  is  no  remainder,  each  time  dividing  the  last  divisor  by 
the  last  remainder.     The  last  divisor  is  the  G.C.D. 

When  there  are  more  than  two  numbers,  find  the  G.C.D. 
of  any  two  of  them,  then  of  that  divisor  and  a  third  7iumber, 
and  so  on  till  all  the  numbers  have  been  used.  The  last  G.C.D. 
is  the  one  required. 

EXAMPLES. 
Find  the  G.C.D.  of 

1.  187  and  153.  9.  3432  and  4760. 

2.  323  and  374.  10.  4939  and  3143. 

3.  434,  539,  and  616.  11.  3696  and  1440. 

4.  1235  and  1495.  12.  9249  and  10920. 

5.  1181  and  2741.  13.  2618,  39039,  and  1771. 

6.  1417  and  1469.  14.  43700  and  9430. 

7.  630,  840,  and  2772.  15.  13860  and  38500. 

8.  13212  and  1841.  16.  17640  and  18375. 

17.  4994,  12485,  and  16117. 

18.  36864,  and  20736. 

19.  156,  585,  442,  and  1287. 

20.  1274,  2002,  2366,  7007,  and  13013. 

Least  Commo^st  Multiple. 

21.  A  multiple  of  a  number  is  any  number  in  which  it  is 
contained  exactly. 

A  common  multiple  of  two  or  more  numbers  is  a  number 
that  will  exactly  contain  eacjj  of  thejft. 


FACTORS.  33 

The  least  common  multiple  of  two  or  more  numbers  is  the 
least  number  that  will  exactly  contain  each  of  them.  For 
example,  48  and  72  are  common  multiples  of  6  and  8,  but  24 
is  the  least  common  multiple. 

"  For  convenience  L.G.M.  is  used  to  represent  the  least 
common  multiple. 

I.   Find  the  L.G.M.  of  36,  42,  and  88. 

36  =  22  X  32.  The  L.C.M.  must  consist 

42  =  2  X  3  X  7.  of  all  the  different  factors 

88  =  2^  X  11.  that  are   in   the   numbers, 

L.C.M.  =  2«  X  32  X  7  X  11  =  5544.       ^"'^  '*'^'  ^'"''"''  ™"«*  ^' 

present  as  many  times  as  it 

is  in  any  one  number.    The  L.C.M.  is  thus  seen  to  be  2'*x3'^x7  xH, 

which  equals  6644. 

To  find  the  L.C.M.  of  two  or  more  numbers,  resolve  the 
numbers  into  their  prims  factors,  and  find  the  product  of  all 
the  different  factors,  taking  each  factor  the  greatest  number  of 
times  it  occurs  in  any  number. 

If  the  numbers  are  prime  to  each  other,  their  product  is 
their  L.C.M. 

The  following  arrangement  of  work  may  be  used : 

2') 36     42     Rf^  Arrange  the  numbers  in  a 

9vr« — 91 — JT  ^^^^  ^^^  divide  by  all  the 

^' —  prime  numbers  that  will  be 

o)J     21     22  contained  in  any  two  of  the 

3        7     22  numbers,  bringing  down  in 

L.C.M.  =  22  X  32  X  7  X  22  =  5544.       ^^"^  ^"'^  ^^^"'"^  *'^^  quotients 

and  the  numbers  that  can- 
not be  divided,  and  so  on  until  the  numbers  in  the  line  are  prime  to 
each  other.  The  product  of  the  divisors  and  the  numbers  in  the  last 
line  is  the  L.C.M. 

EXAMPLES. 
Find  the  L.C.M.  of 

1.  15,  18,  and  35.         3.  36,  48,  and  72. 

2.  20,  24,  and  36.         4.  10, 14, 15, 21, 3(3,  and  42. 


34  ARITHMETIC. 

6.  48,  98,  21,  and  27.  14.  84,  126,  and  140. 

6.  18,  32,  48,  and  52.  15.  156,  234,  and  260. 

7.  48,  26,  72,  and  24.  16.  105,  476,  and  306. 

8.  21,  36,  50,  and  64.  17.  144  and  780. 

9.  14,  36,  108,  and  144.  18.  240,  560,  and  616. 

10.  72,  80,  84,  and  96.  19.  740,  333,  and  296. 

11.  91,  52,  39,  28,  and  21.  20.  945  and  1485. 

12.  3,  91,  78,  182,  and  231.  21.  936  and  2925. 

13.  108,  217,  54,  and  31.  22.  504,  924,  and  2184. 

23.   1209,  1885,  2457,  2691,  and  2717. 

24.  Find  the  L.C.M.  of  the  nine  digits. 

25.  Find  the  L.C.M.  of  the  even  numbers  from  10  to  20 
inclusive. 

26.  What  is  the  shortest  length  that  can  be  measured  by 
either  of  three  measures,  which  are  respectively  9,  15,  and 
24  inches  long  ? 

27.  Find  the  contents  of  the  smallest  cistern  that  can  be 
exactly  measured  by  either  one  of  three  casks  containing 
respectively  18,  25,  and  30  gallons. 

28.  What  is  the  width  of  the  narrowest  walk  that  can  be 
paved  with  blocks  each  12  inches  long  and  15  inches  wide, 
allowing  the  blocks  to  run  either  lengthwise  or  across  the 
walk? 

29.  What  is  the  smallest  sum  of  money  that  can  be  made 
up  either  of  2-cent,  of  3-cent,  of  5-cent,  of  10-cent,  or  of  25- 
cent  pieces  ? 

30.  Four  boys  start  together  to  run  around  a  square ;  the 
first  can  run  around  in  12  minutes,  the  second  in  15  minutes, 
the  third  in  16  nrinutes,  and  the  fourth  in  18  minutes ;  how 
long  will  it  be  before  they  all  meet  at  the  starting-point  ? 


FACTORS..  35 

22.  When  the  numbers  cannot  be  factored  easily,  find 
the  G.C.D.  of  two  or  more  of  the  numbers,  and  use  it  as  a 
divisor  in  the  first  line,  and  then  proceed  as  before. 

I.   Find  the  L.C.M.  of  368,  483,  and  532. 

368)483(1  23)368(16  23)483(21  23)532(23 

368  23_  46_  46_ 

115)368(3  138  23  72 

345  138  23  69 

23)115(5  3 

115 

23)^68^8^432  J^^^^a"  .e  .f  Vh" 

divide  all  the  numbers  by 

23.    It  is  contained  in  the 

first  two, but  not  in  tlie  third. 

4  3        19  However,  the  numbers  are 

L.C.M.  =  23  X  22  X  7  X  4  X  3  X  19       """"^  '"*  '"'''^^  simplified  that 
__  145832  *he    method    as   previously 

given  can  be  applied. 

When  there  are  but  two  numbers,  the  L.C.M.  can  be  found 
by  dividing  one  number  by  the  G.C.D.  and  multiplying  the 
quotient  by  the  other  number. 


2)  16 

21 

532 

2)  8 

21 

266 

7)  4 

21 

133 

Find  the  L.C.M.  of 

EXAMPLES. 

1. 

187  and  153. 

9. 

1217,  1422,  and  1611. 

2. 

391  and  493. 

10. 

3150  and  2310. 

3. 

209,  247,  and  253. 

11. 

9249  and  10920. 

4. 

187,  539,  and  847. 

12. 

4939  and  3143. 

5. 

630,  840,  and  2772. 

13. 

2618,  39039,  and  1771. 

6. 

1417  and  1469. 

14. 

43700  and  9430. 

7. 

iOll,  1685,  and  2359. 

15. 

13860  and  38500. 

8. 

1517  and  1763. 

16. 

2520,  2772,  and  30888. 

17.  17640  and  18375. 

18.  340200,  583200,  and  2268000. 


36  ARITHMETIC. 


.     Cancellation. 

23.  Division  may  be  indicated  by  writing  the  dividend 
above  a  line  and  the  divisor  below  the  same  line.  For  ex- 
ample, \2_  means  42  divided  by  6.  This  method  of  indicat- 
ing division  is  commonly  used  when  there  are  several  factors 
in  either  dividend  or  divisor.  Such  an  example  in  division 
can  be  simplified  by  striking  out,  or  cancelling,  like  factors 
in  dividend  and  divisor.  This  does  not  affect  the  result,  be- 
cause when  both  dividend  and  divisor  are  divided  by  the 
same  number  the  quotient  remains  the  same. 

I.  Find  the  value  of  ^  X  5  x  30  x  12 

20  X  7  X  6 

5         3  Cancel  7  and  7.     Then  cancel  5  in  the 

J  X  p  X  TJf!)  X  7^  __  -j^K        dividend  and  20  in  the  divisor,  writing  4 

.  ^0  X  y  X  ^  '       below  the  20,  as  20  divided  by  5  equals  4. 

^  Then  cancel  4  and  12,  writing  3  above  the 

12 ;  cancel  6  and  30,  writing  5  above  the  30.     The  result  is  5  X  3,  which 

equals  15. 

II.  How  many  pounds  of  sugar  worth  9  cents  a  pound 
must  be  given  in  exchange  for  18  dozen  of  eggs  worth  1^ 
cents  a  dozen  ? 

fy  The  value  of  the  eggs  is  18  X 16  cents, 

^g,        -tn  and  as  many  pounds  of  sugar  can  be  ob- 

^      =  32  pounds.  tained  as  9  is  contained  times  in  18  X  16. 

r  Simplify  by  cancellation. 


EXAMPLES. 


Find  the  value  of 


1  5  X  8  X  3  X  16  3    9  X  25  X  64 

*  8x15x4    '  *  5x18x32 

2  11x33x21x13  ^    20  X  56  X  12 

*  13x7x11x3"  '21x10x8' 


FACTORS.  37 


792  rj    54x84x99 


11x4x9  9x22x63 

g    108  X  132  g    57  X  119  X  16 

*    99x144*  '17x12x19* 

9.  How  many  yards  of  cloth  worth  22  cents  a  yard  must 
be  given  in  exchange  for  11  bushels  of  potatoes  worth  60 
cents  a  bushel  ? 

10.  If  50  oranges  cost  75  cents,  find  the  cost  of  30 
oranges. 

11.  How  many  barrels  of  flour  can  be  bought  for  $247, 
when  8  barrels  cost  $52  ? 

12.  If  16  men  can  dig  a  ditch  in  21  days,  how  long  will 
it  take  24  men  ? 

13.  If  the  work  of  11  men  equals  the  work  of  17  boys, 
how  many  men's  work  will  equal  the  work  of  68  boys  ? 

14.  Allowing  17  bushels  of  wheat  to  make  4  barrels  of 
flour,  how  many  bushels  will  be  necessary  to  make  68 
barrels  ? 

15.  When  15  barrels  of  pork,  each  containing  200  pounds, 
are  worth  f  250,  find  the  value  of  60  pounds. 

16.  How  many  dresses,  each  containing  16  yards,  can  be 
made  from  20  pieces  of  cloth,  52  yards  in  each  piece  ? 

17.  If  54  men  can  build  a  wall  in  35  days,  working  10 
hours  a  day,  how  many  men  will  be  necessary  to  build  it 
in  15  days,  working  9  hours  a  day  ? 

18.  A  gardener  sells  75  crates  of  berries,  24  boxes  in  a 
crate,  at  8  cents  a  box,  and  receives  in  return  12  rolls  of 
matting,  40  yards  in  a  roll ;  find  the  price  of  the  matting  a 
yard. 


38  ABITHMETIC. 


CHAPTER  IV. 

COMMON  FRACTIONS. 

24.  A  unit  may  be  divided  into  any  number  of  equal 
parts,  and  any  number  of  these  parts  may  be  taken  to- 
gether. For  example,  -f-  means  that  the  unit  is  divided  into 
seven  equal  parts,  of  which  five  are  taken;  it  is  read  jive 
sevenths.  As  has  been  stated  in  §  23,  ^  also  means  five 
divided  by  seven,  but  these  two  meanings  are  really  the 
same,  because  the  quotient  arising  from  dividing  five  by 
seven  is  five  sevenths. 

An  expression  used  to  denote  one  or  more  of  the  equal 
parts  of  a  unit  is  called  a  fraction.  When  it  is  represented 
by  two  numbers,  one  written  above  the  other  with  a  divid- 
ing line  between,  it  is  called  a  common  fraction,  or  vulgar 
fraction.  When  it  is  represented  by  figures  at  the  right  of 
the  decimal  point,  as  shown  in  Chapter  II.,  it  is  called  a 
decimal  fraction. 

In  common  fractions  the  number  below  the  line,  which 
shows  into  how  many  equal  parts  the  unit  is  divided,  is 
called  the  denominator.  Tho  number  above  the  line,  which 
shows  how  many  of  the  equal  parts  are  taken,  is  called  the 
numerator.  The  two  numbers  are  called  the  terms  of  the 
fraction. 

A  proper  fraction  is  one  whose  numerator  is  less  than  its 
denominator ;  as  f . 

An  improper  fraction  is  one  whose  numerator  is  equal 
to  or  greater  than  its  denominator;  as  f,  ^.  When  the 
numerator  is  greater  than  the  denominator,  more  than  one 
unit  must  be  divided  into  equal  parts ;  for  example,  ^  means 


COMMON  FK ACTIONS.  39 

that  three  or  more  units  have  each  been  divided  into  eight 
equal  parts,  and  nineteen  of  these  parts  are  taken. 

A  mixed  number  is  an  integer  and  a  fraction  expressed 
together ;  as  6^-^,  which  is  read  six  and  seven  fifteenths. 

A  compound  fraction  is  a  fraction  of  a  fraction ;  as  J  of  -^. 

A  complex  fraction  is  one  which  has  a  fraction  in  one  or 

both  of  its  terms;  as  -,   -^ . 

The  reciprocal  of  a  number  is  the  quotient  arising  from 
dividing  1  l)y  that  number.     For  example,  the  reciprocal  of 

7  is  f 

25.  Since  a  fraction  is  an  expression  of  division,  tlu;  last 
three  principles  of  §  4  may  be  restated  for  fractions  as 
follows : 

(1)  Multiplying  the  numerator  or  dividing  the  denomi- 
nator by  a  number  multiplies  the  fraction  by  the  same 
number. 

{2)  Dividing  the  numerator  or  multiplying  the  denomi- 
nator by  a  number  divides  the  fraction  by  the  same  number. 

(3)  Multiplying  or  dividing  both  numerator  and  denom- 
inator by  the  same  number  does  not  change  the,  value  of  the 
fraction. 

Eeduction  of  Fractions  to  Lowest  Terms. 

26.  A  fraction  is  in  its  lowest  terms  when  the  numerator 
and  denominator  are  prime  to  each  other. 

1.   Reduce  ^^^  to  its  lowest  terms. 

2^5  21 3  Since  dividing  both  numerator  and  denominator  by 

^iT  6  3  y  ^\^Q  same  number  does  not  change  the  value  of  the 
fraction,  we  can  divide  both  terms  by  5  and  thus  obtain  ||  for  a  value 
of  the  fraction  in  lower  terms.  Then  divide  both  terms  by  9,  and  we 
obtain  f .  Since  3  and  7  are  prime  to  each  other,  the  fraction  is  in  its 
lowest  terms. 


40  ARITHMETIC. 

To  reduce  a  fraction  to  its  lowest  terms,  divide  numerator 
and  denominator'  successively  hy  their  common  factors.  Since 
the  product  of  all  the  common  factors  is  the  G.C.D.,  in  ail 
cases  where  the  common  factors  are  not  easily  seen,  divide 
numerator  and  denominator  hy  their  G.G.D. 

EXAMPLES. 

Beduce  the  following  fractions  to  their  lowest  terms . 

14  2  »       625  IK       4  3  4  3 


TW 


392 


2.  ^.  9. 

3.  ff.  10..  m 

4.  i^.  11.   Ill 

5.  ^.  12.  m 

6.  iff.  13-    t\%V  20. 

7.  |M.  14.   -MA.       •  21. 


"JT5T- 

16-  im- 

17-  im- 

18-  Mm 

1^-     Hf'ST- 


23820 


1  84800 


22.    Ascertain  whether  the  fraction  -exWr  ^^  "^  ^*^  lowest 
terms  or  not,  and  explain  the  process  you  employ. 

Reduction  of  Improper  Fractions  to  AViiole  or 
Mixed  Numbers. 

27.   I.   Reduce  -L3^  to  a  whole  number. 
13^  =  11.  W-  is  the  same  as  132  h-  12,  which  equals  11. 

II.    Reduce  i^-  to  a  mixed  number. 

J.J  ^^  g    ^^3      -lyY"  is  the  same  as  141 -f- 12,  which  equals  lly'^ 

T^  12  ^'     and  this  reduces  to  11|. 

EXAMPLES. 

Reduce  to  whole  or  mixed  numbers 

1.  ¥•  3.  -VV-.  5.  -w. 

2.  ||.  4.   y/.  6.   ^. 


COMMON    FRACTIONS.  41 

8.  ^^.  11.  le^.         14.  i^. 

9.   V/.  12.   i-fp.  15.   -V//- 

Reduction   of   Whole   or   Mixed   Numbers  to 
Improper  Fractions. 

28.   I.   Reduce  8  to  sevenths. 

8  =  Af-.         Since  there  are  7  sevenths  in  1,  in  8  there  are  8  times 
7  sevenths,  whicli  equals  ^y". 

A  whole  number  may  be  written  as  a  fraction  with  1 
for  the  denominator.     For  example,  13  =  X^. 

II.   Reduce  12|^  to  an  improper  fraction. 
12 J  =  108  +  7  =  i^.        12  =  101 ;  adding |  to  this,  the  result  is  i^; 

EXAMPLES. 

Reduce  to  improper  fractions 

4.  12^.  7.   9|i. 

5.  16,^.  8.   12tt. 

6.  21,^.  9.   15^. 

10.  Reduce  9  to  eighths. 

11.  Reduce  12  to  elevenths. 

12.  Reduce  19  to  thirteenths. 

13.  Reduce  25  to  fifteenths. 

14.  Reduce  42  to  twenty-fourths. 

Least  Common  Denominator. 

2^.  When  several  fractions  have  the  same  denominator. 
they  are  said  to  have  a  common  denominator.  It  is  always 
possible  to  reduce  two  or  more  fractions  to  equivalent  frac- 


1. 

3|. 

2. 

7*. 

3. 

10i|. 

42  ARITHMETIC. 

tions  having  a  common  denominator ;  but  the  most  useful 
common  denominator  is  the  least  common  multiple  of  the 
denominators,  which  is  known  as  the  least  common  denomi- 
nator. For  convenience  L.C.D.  is  used  to  represent  the 
least  common  denominator. 

I.   Eeduce  |,  J,  and  \^  to  equivalent  fractions  having  the 
L.C.D. 
5  _  _5^   _  30,         We  find  the  L.C.M.  of  6,  9,  and  12  to  be  36.    6 
7 7x4    28       must  be  multiplied  by  6  to  obtain  36;   hence  5 

must  be  multiplied  by  6  in  order  to  keep  the  frac- 

11  --11  X3.—  33  I'  J  i- 

12  3  6         3  6"      tion  of  the  same  value,  because  multiplying  both 

numerator  and  denominator  by  the  same  number 
does  not  change  the  value  of  the  fraction.  A  similar  process  is  applied 
to  the  other  two  fractions. 

To  reduce  fractions  to  equivalent  fractions  having  the 
L.C.D.,  in  each  fraction  multiply  both  terms  by  the  quotient 
arising  from  dividing  the  L.C.D.  by  the  denominator. 

Note.  Fractions  should  always  be  in  their  lowest  terms  before  find- 
ing the  L.C.D. 

EXAMPLES. 
Reduce  to  equivalent  fractions  having  the  L.C.D. 

1.  1  f,  and  f .  8.   1  \,  \,  I,  and  \. 

2.  I,  f,andTV  9.   i,  |,  ^V.  and  |i. 

3.  if,  ^2.  and  If.  10.   f ,  A,  2\.  and  ^3^. 

4.  f,T\,andif.  11.   A.  A.  i*.  and  if . 

5.  I,  I,  and  ^.  12.   ^-^^  ||,  _2_  and  ^. 

6.  A.  \i.  and  ^.  13.   A,  \^,  ^,  ^,  and  ^. 

7.  |,A,and3-V  14.   A,  H.  H.  If .  and  ||. 

15.  t\,  II,  ^3_  _9_9_  100^  and  ^\. 

16.  ih  U^  n,  T^A.  T^B^.  and  AV 


COMMON   FRACTIONS.  48 


Addition  of  Fractions. 

30.   I.   Find  the  sum  of  3^,  3^,  and  ^. 

2  j^  5  ,  g  _  2-^5+8  —  15  Quantities  to  be  added  together 
tV-f-TV  +  lT—  "tY""— T7-  jnust  ^g  ^f  the  same  kind.  Since 
the  denominators  are  alike,  we  have  merely  to  add  the  numerators. 

II.  Find  the  sum  of  f,  f,  and  ^^. 

When  the  denominators  are  unlike,  the  fractions  must  first  be 
reduced  to  equivalent  fractions  having  the  L.C.D.,  and  then  added. 

III.  Find  the  sum  of  2\,  3|,  6|^,  and  ^. 

2i  +  3|  +  6ii  +  A  =  lU^±&|F±i^  =  lift  =  12H  =  12i. 

The  sum  of  the  whole  numbers  is  11,  and  the  sum  of  the  fractions 
is  II;  the  two  sums  taken  together  equal  12§J,  which  reduced  to  its 
lowest  terras  becomes  12|. 

To  add  fractions,  reduce  the  fractions  to  equivalent  fractions 
having  the  L.C.D.,  and  ivrite  the  sum  of  the  numerators  over  the 
L.C.D.  When  there  are  mixed  numbers,  add  the  whole 
numbers  and  fractions  separately,  and  combine  the  results. 
Improper  fractions  should  be  reduced  to  mixed  numbers  be- 
fore adding. 

EXAMPLES. 

Find  the  sum  of 

1.  4,f,and-^.  7.  f,|,H.and4f. 

2.  t,  I,  and  f  8.  |,  A,  {i,  and  Jf. 

3.  i,^,2indj\.  9.  1%,  H.  A.  A.  and  ^. 

4.  A  s%  A  and  H-  10.  i,  ^,  ^,  and  ^. 

5.  f,|,f,and|i  11.  I,  I,  2^3^,  and  S^V 

6.  I,  I,  J,  and  ^^.  12.  5f ,  3 A,  2^,  H>  and  21 


44  AHITHMETIC. " 

13.  H.  4i,  \%  and  1,%.  19.  5^,  2f ,  ^,  and  ^f .         . 

14.  5f ,  Jf ,  I,  and  if .  20.  5|i,  f,  3^,  and  1^^. 

15.  2f ,  H,  3i,  and  f  21.  «,  ^V,  and  1|S|. 

16.  i,  A.  tV.  and  4J.  ■    22.  4^,  7^,  \\,  |f,  and  Iff. 
17-  5f  If,  il,  and  3^.  23.  |,  A,  Iff,  and  ff . 

18.   I,  il,  11  A.  and  iJ.  24.   if,  ^,  and  2^^. 

25.  12^,  13^^,  17|,  and  J^^^. 

26.  18  A,  12i|,  104,1^,  and  29^. 

Subtraction  of  Fractions. 
31.   I.    Subtract  I  from  jL. 

The  fractions  must  first  be  reduced  to 
^  —  1^  =  ^  ^^^  ^  =  g^g-.  equivalent  fractions  having  the  L.C.D., 

and  then  subtracted. 

II.  Subtract  42^  from  7yV 

The  difference  between  the  whole  num- 

7  9       A  5  Q 27-10 Q17    ^^^^  ^^  ^>  ^"^  ^^^^  difference  between  the 

^^        2^        ~T^  ¥Y'  fractions    is    ^|;   these   two   differences 

taken  together  equal  Z\\. 

III.  Subtract  5f  from  llf . 

112-5|  =  68^  =  5^^#i  =  5i4.         '^^  ''^''"''*  subtract  fi 
'  *  ^  **  2  ^  2  ®        from  /g,  so  we  take  1  from 

0,  which  makes  f  f  to  be  added  to  aV  J  the  example  thus  becomes  S^-Vg— , 
which  equals  h\\. 

To  subtract  fractions,  reduce  the  fractions  to  equivalent 
fractions  having  the  L.G.D.,  and  ivrite  the  difference  between 
the  numerators  over  the  L.C.D.  When  there  are  mixed  num- 
bers, subtract  the  whole  numbers  and  fractions  separately, 
and  combine  the  results.  Improper  fractions  should  be 
reduced  to  mixed  numbers  before  subtracting. 


COMMON  FRACTIONS.  45 

EXAMPLES. 
Find  the  difference  between 

1.  I  and  f .  12.  3j\  and  2^. 

2.  ^andf  13.  9yV  and  8f 

3.  |andf  14.  4f  and  3f 

4.  -y-and^?^.  15.  15^2^  and  12^. 

5.  T^and^^.  16.  36y9^  and  f 

6.  3^and^.  17.  7^  and  6^. 

7.  ^and/^.  18.  20^  and  19^^. 

8.  i^and^.  19.  4/^  and  2|^. 

9.  T^^andii^.  20.  19^  and  9^. 

10.  TfJ^and^Hr.  21.   20^  and  15^- 

11.  18f  andl5i.  22.   lO^i^  and  9^- 

Addition  and  Subtraction  of  Fractions  Combined, 
32.   I.   Simplify  8/^ -2J-3J  + 635^ -5{. 
8A  +  6A  =  14it^  =  14ff. 
2i  +  3i  4-  5|  =  10^i:^+-i^  =  10|i  =  llif . 
14i|  -  llif  =  31^:^  =  211^  =  2}f  =  2H. 

The  three  terms  preceded  by  the  minus  sign  are  to  be  subtracted 
from  the  sum  of  the  remaining  terms.  Subtracting  the  sum  of  these 
terms  gives  the  same  result  as  if  they  were  subtracted  separately. 

EXAMPLES. 

Simplify 

1.  9j_6A  +  2|-H-4A. 

2.  7i-3j%-TV-i  +  H- 

a   28^^-7^  +  16^  +  41-14^. 


46  ARITHMETIC. 

5.  15-2H-4A-6f-A. 

6.  7|--6f  +  5f-4f+3|-2|  +  lf 

7.  40ff- 3^^-5^-14^- 8 +  12101 -16^. 

8.  13|-2,^-6A  +  3-l,%  +  8f-ff-10Jf. 

9.  Add  together  f,  ^,  and  y\,  and  from  their  sum  sub- 
tract ^. 

10.  -^^  of  a  pole  is  in  the  mud,  -^  of  it  is  in  the  water, 
and  the  rest  of  it  is  in  the  air ;  what  part  of  it  is  in  the  air  ? 

11.  In  a  school  J  of  the  scholars  are  Germans,  |  Irish,  ^ 
English,  Jg-  Swedes,  and  the  remainder  Americans;  what 
part  of  the  scholars  are  Americans  ? 

12.  A  man  performs  a  journey  of  84 J  miles  in  five  days. 
He  travels  12|  miles  the  first  day,  18J  miles  the  second 
day,  16f  miles  the  third  day,  and  20|-  miles  the  fourth  day ; 
how  far  does  he  travel  the  fifth  day  ? 

13.  A  merchant  bought  two  pieces  of  cloth  containing 
46|-  yards  and  53|-  yards  respectively.  He  sold  12|  yards, 
15|  yards,  18|-  yards,  and  14y^2  yards;  how  many  yards 
were  there  remaining  ? 

14.  A  painter  receives  $15  for  painting  a  room.  He  ex- 
pends $6\  for  labor,  $4y^^  for  paint,  and  $2-^-^  for  varnish ; 
find  the  amount  he  gained. 


Multiplication  of  Fractions. 

33.   I.   Multiply  I- by  3. 

Since  multiplying  the  numerator  by  a  number  multi- 
^  X  3  =  y.       plies  the  fraction  by  the  same  number,  we  multiply  the 
numerator  by  3,  and  obtain  f  as  the  result. 


•COMMON  FRACTIONS.  47 

II.  Multiply  fxf 

If  we  multiply  the  numerators  together,  we  obtain 

4  X  #  =  f  §•     5X4.     Since  dividing  either  multiplicand  or  multiplier 

by  a  number  divides  the  product  by  the  same  number, 
if  we  divide  one  by  7  and  the  other  by  9,  we  divide  the  product  by  7  X  9. 

5  divided  by  7  equals  f ,  and  4  divided  by  9  equals  f ;  hence  the  product 
of  f  and  I  is  5x4  divided  by  7  XO,  which  may  be  written  ^^,  and  is 
equal  to  f  §. 

III.  Multiply  I  by  H- 

g      *2      4  f  XT5  =  fx'3  5-      '^^^  principle  of  cancellation  can 

^  X  —  =  — •  then  be  applied,  but  it  gives  the  same  result  to  apply 
3      7  the  cancellation  at  once  to  the  example. 

IV.  Multiply  together  f ,  ^,  and  5\. 

ft      M     il      ^'i  Reduce  5}  to  an  improper 

^  X  ii  X  6^  =  ^  X  J^  X  ^  =  Jl  =  2  r^j.  fraction,  and  proceed  as  in  the 

9      do                9      ^p       4       15  J,                    , 

3      5  precedmg  example. 

To  find  the  product  of  a  whole  number  and  a  fraction, 
ivriie  the  product  of  the  whole  number  and  the  numerator  over 
the  denominator.      • 

To  find  the  product  of  several  fractions,  write  the  product  of 
the  numerators  over  the  product  of  the  denominators,  first  can- 
celling the  factors  common  to  a  numerator  and  denominator. 
Mixed  numbers  should  be  reduced  to  improper  fractions 
before  multiplying,  and  a  whole  number  should  be  treated 
as  a  numerator. 

Compound  fractions  are  simplified  by  multiplying  to- 
gether the  simple  fractions.     For  example,  |  of  |-  =  f  x  f . 


^.    ,  .,             ,           .      EXAMPLES. 
Find  the  product  of 

1.   j\Xi.                     4.   iof^. 

7. 

H  X  H- 

2.  48xA-                 5.   ix3i. 

8. 

Hx6|. 

3.  ^off                    6.  e^xff. 

9. 

^x^. 

48  ARITHMETIC. 

10.  8^x22V  .      18.  2ixl^xl^xA- 

11.  I  off  off  19.  ^ot3^xlj\Xj\. 

12.  AxAxil.  20.  5,VXT%of||oflJ,. 

13.  l^XfiofTf  21.  i|-x2|XTteXlA- 

14.  J  X  f  X  If  X  If  22.  3f  X  iJ  X  2ii  X  If  of  14. 

15.  101x21x^x3^.  23.  lAx2ifx2i|xlxVxfi. 

16.  Hx3fx3|Xii'  '24.  lAxl7Jxlifxl9Vx2A. 

17.  2ix4fx5fxf  25.  |x||xAx7ixl6eV 

34.  In  finding  the  product  of  a  mixed  number  and  a 
whole  number,  or  of  two  mixed  numbers,  we  can  use  another 
method,  which  is  particularly  useful  when  the  integral 
parts  of  the  mixed  numbers  are  large  numbers. 

I.   Multiply  23  by  6^. 

23 
Q  7  Multiply  23  by  7  and  divide  the  result  by  11,  which  is 

il'ilfil  *'^^  same  as  multiplying  by  y\ ;  this  gives  14 j7p     Then 

multiply  23  by  6,  and  add  138  thus  obtained  to  14^7^. 
J4yy       fjjjg  gjj^jj.^  product  is  152^.        , 
loo 

152^ 


II.   Multiply  18| 

by8i. 

18| 

8i 

.tV        |xi= 

=  tV;  18Xi  =  4i;  1X8  = 

=  3^;  18X8  = 

=  144.  T 

^^          sum  of  these  partial  products  is 

151|. 

H 

144 

151f 

EXAMPLES. 

Mnd  the  product 

of 

1.   41x3f 

3.   25x6f 

5. 

29  X  21tV 

2.  23x6J. 

4.  32x10jV 

6. 

18  X  11^. 

*  COMMON   FRACTIONS.  49 

7.  8ix6J.  11.  14|x8|.  15.  31ifxl4|. 

8.  8fx5|.  12.  12tVx43^.  16.  G6|  x  37J. 

9.  22ix8f.  13.  18|xl2f.  17.  112^  x  31J. 
10.  lly^xSf  14.  25^x16^.  18.  168ix83i. 

Division  of  Er actions. 

35.  I.   Divide  I  by  f 

5.6  —  5  1—  M.  ^  divided  by  1  equals  f .  If  we  diyide  the 
^  '  T'~"^  b  -~  Ti'  divisor  by  7,  we  multiply  the  quotient  by  7 ; 
hence  |  divided  by  }  equals  ^§^.  If  we  now  multiply  the  divisor  by 
(),  we  divide  the  quotient  by  6 ;  hence  ^  divided  by  f  equals  f^.  This 
is  the  same  as  |  X  |,  which  equals  ||. 

To  divide  a  fraction  by  a  fraction,  inveii,  the  divisor  and 
proceed  as  in  multiplication. 

II.   Divide  A'of  2^.  by  -^  of  5f 


;2 

2                         ? 

_  3 
"14* 

When  the  divisor  contains  more  than  o 
factor,  each  factor  should  be  inverted. 

EXAMPLES. 

Find  the  quotient  of 

1-    l^f. 

9. 

H^H- 

17. 

10^^13. 

2.    18-1- 

10. 

4|^6|. 

18. 

3H-^iff. 

3.   3-^^15. 

11. 

100 -4f. 

19. 

3A-2«. 

4.    H-^ff. 

12. 

4fJ^10^. 

20. 

m-^m- 

5.   5^^  If. 

13. 

tVo^A- 

21. 

23-T    •    241- 

6.   If  ^21 

14. 

m^m- 

22. 

lOM^irfr- 

7.   5^-^21. 

15. 

u^n- 

23. 

5f-^^- 

8.    161 --42. 

16. 

lOi^iH- 

24. 

7*^31^. 

50  ARITHMETIC. 

25.  foflH-fof^.  29.  3^of2i-JVof  5i|. 

26.  J  of  l|  -  A  of  If.  30.  i  of  f  of  2i  -  f^  of  If 

27.  I  of  91  -- 1^  of  637.  31.  f  of  ^\  --  3%  of  3^  of  4^. 

28.  I  of  7i-|  of  11^.  32.  I  of  I  of  A^f  ^^  i  of  I- 

36.    When  the  divisor  is  either  a  whole  or  a  mixed  num- 
ber, a  different  method  may  often  be  used  to  advantage. 

I.  Divide  29f  by  4. 

^/-^"t  29  divided  by  4  equals  7  with  a  remainder  of  1.     If 

7^      divided  by  4  equals  V  X5,  which  equals  |f. 

II.  Divide  52|  by  3|. 

"^^z   A-"^  Since  multiplying  both  dividend  and   divisor  by  the 

^          ^  same  number  does  not  affect  the  quotient,  we  can  multi- 

11)157^  ply  both  dividend  and  divisor  by  3,  the  denominator  of 

14|4.  the  divisor,  and  then  proceed  as  in  the  preceding  example, 

EXAMPLES. 
Find  the  quotient  of 


1. 

22i-r-3. 

7. 

156J-25. 

13. 

481-51 

2. 

19|  --  5. 

8. 

128^ --30. 

14. 

561-^81 

3. 

283-9^ -7. 

9. 

22 -6|. 

15. 

104yi--^8f 

4. 

33^-12. 

10. 

21  -  5|. 

16. 

906-^111 

5. 

44J  :  15. 

11. 

64-9^. 

17. 

115|-21f 

6. 

87-5- -J- 21. 

12. 

13t^2i. 

18. 

402^-30^. 

Short  Methods  of  Multiplication  and  Division. 

37.  Any  exact  fractional  part  of  a  number  is  called  an 
aliquot  part  of  that  number.  For  example,  2,  2^,  3^,  and  5 
are  aliquot  parts  of  10. 


COMMON   FRACTIONS.  51 

To  multiply  by  aliquot  parts  of  10,  100,  1000,  etc.,  multi- 
ply hy  10,  100,  1000,  etc.,  as  the  case  may  j-equire,  and  then 
find  the  required  part.  For  example,  since  16J  =  ^  of  100, 
24  X  16|  =  I-  of  2400  =  400. 

To  divide  by  aliquot  parts  of  10,  100,  1000,  etc.,  divide 
by  10,  100,  1000,  etc.,  as  the  case  may  require,  and  then 
multiply  hy  the  denominator  of  the  fraction  expressing  the 
aliquot  part.  For  example,  since  12^  =  i  of  100,  225  ^  12^ 
=  2.25x8  =  18. 

To  multiply  by  a  number  a  little  less  than  10,  100,  1000, 
etc.,  multiply  by  10,  100,  1000,  etc.,  and  from  the  product 
subtract  the  product  of  the  multiplicand  by  the  difference  be- 
tween the  midtiplier  and  10,  100,  1000,  etc.,  as  the  case  may 
require.  For  example,  184  x  99  =  18400  -  184  =  18216  ; 
184  X  98  =  18400  -  368  =  18032. 

EXAMPLES. 

1.  Multiply  423  by  5.  14.  Divide  11150  by  25. 

2.  Multiply  2918  by  2f  15.  Divide  2700  by  16f . 

3.  Multiply  57162  by  3^.  16.  Divide  42125  by  12f 

4.  Multiply  3143  by  25.  17.  Divide  1172  by  331- 

5.  Multiply  4890  by  16J.  18.  Divide  87320  by  250. 

6.  Multiply  12792  by  33J.  19.  Divide  1183^  by  166J. 

7.  Multiply  804320  by  12^.  20.  Divide  33625  by  125. 

8.  Multiply  84322  by  250.  21.  Multiply  64  by  9. 

9.  Multiply  7614  by  125.  22.  Multiply  82  by  99. 

10.  Multiply  5436  by  166|.  23.  Multiply  127  by  999. 

11.  Divide  7165  by  5.  24.  Multiply  7342  by  9999. 

12.  Divide  8775  by  2^.  25.  Multiply  138  by  98. 

13.  Divide  876|  by  3J.  26.  Multiply  72  by  997. 


52  ARITHMETIC. 


Complex  Fractions. 

5  ^ 
38.  I.   Reduce  -^  to  a  simple  fraction. 

5  A  complex  fraction  indicates  that  the  nu- 

5/^  _  ^  2  _  5            merator  is  to  be  divided  by  the  denominator. 

6^      X^  X^     6            Hence  we  perform  the  division  by  inverting  the 

^  divisor  and  proceeding  as  in  multiplication. 

3 

II.   Reduce  5  to  a  simple  fraction. 
5 

a        o  III  many  cases    the  simplest  solution  is  to 

^  =  —  •  multiply  both  numerator  and  denominator  by 

their  least  common  denominator.     In  this  ex- 
ample we  multiply  both  terms  by  4. 


III.   Reduce  -5 — ^  to  a  simple  fraction. 


6+-t 


3i-2J  =  ll^=9_2  =  7 

The  numerator  and  denomi- 
2f  +  If  =  3-^  =  3f  =  4|  =  4|-.  nator  must  each  be  simplified, 
J_  _  7  w  2  _  _Z..  ^^^^  ^^^"  ^^®  ^'^'^  proceed  as  in 

4|     ^     9     36  the  preceding  examples. 

4 

EXAMPLES. 

Reduce  the  following  complex  fractions  to  simple  frac- 
tions : 


2. 


4.    ^ 


6i 

3f 

5fi-5i 

10 

6. 

18^ 

10. 

4]t-2i 

16| 

1  of  1  of  1 

6i-2| 

9f 
2* 

7. 

i  of  f  of  7f 
19A 

11. 

2i  +  2f 
4f-3f 

6i 
33i" 

8. 

2i  +  5i 
i 

12. 

3i-2i. 

COMMON  f'l^ACTIO^fs.  53 


13. 

3|-2i 
H  +  3^ 

14. 

3i  +  ii-| 

6*-fxi 

15 

2|^4x2 

.-|.5 

16 

4*  +  24-^| 

6i-l|x| 

17 

lT^_9}  +  4^ 

|x9i^ 

19. 


4* 


'^^n 


20  ItziM. 

45      149 

U      20 

21  i  +  f  +  i-A 

3t-2i 


22. 


23. 


(3i-2i)^l| 
l|  +  2i 

f  off  ^f  of  A 
7i-Hx5f  • 


18     (4i  +  7i)^3i  24       ^  +  i  +  i 

'      "      ""      ^-  '    i  +  i  +  - 

2i^3i^4i 


|x2^x5i  •    }_^1__^X_ 


39.  When  in  a  series  of  fractions  we  have  only  the  signs 
of  multiplication  and  division,  one  operation  is  sufficient  to 
obtain  the  result. 

.   I.    Simplify    tQ^TT       jj. 
5f-2ii     12f 

5*  -  2H  =  3^^^  =  3A  =  3f  •  We  must  first  perform  the 

f  of  |-^        1^       f  of  ^      12f  subtraction  in  the  denominator 

3i        ~  12^ ~~      3I  II  of  the  first  fraction.     We  can 

3  then  invert  the  second  fraction 

—  §X  —  X— X  —  X  —  =  -•  *"^   obtain  the  result   by   one 

^     m     %'S      4      ^^4  process  of  cancellation. 


64  ARITHMETIC. 

EXAMPLES. 

112  Q 

1.  Find  in  its  simplest  form  the  value  of  — ^  -e — 

^  12f     9 

2.  Multiply  ||  by  ^. 

3.  Multiply  ||  by  3-«j  of  2i. 

4.  Multiply  f  of  p  by  I  of  i. 

5.  Multiply  I  of  y_  of  4i  by  :y^%^' 

6.  What  is  the  product  of  f  of  ^^  of  15  and  |f  of  llf  ? 

7.  Reduce  to  its  simplest  form  A  of  \8-  of  31  ^  — ?ii— , 

8.  Divide  I  by  il. 

6  "^F 

9.  Divide  |i  x  72|  by  |  of  f  of  9|. 

10.  Divide  iof  12|-  by  X  of  8f. 

TT  '^^ 

11.  Divide  A  of  ^  of  8i  by  t^f*. 

12.  Divide  -^^  of  ^  of  7^  by  — iM — 

13.  Divide  10  times  f-  of  -il-  of  9A^  by  -t. 

V9     12^      ^y  ^  1\ 

14.  Reduce  -|  x  -|  h- A  to  its  simplest  form. 

^^4         '*t^        ^6 

15.  Reduce     ^  ^  ^,  -=-  ,    \       to  its  simplest  form. 


COMMON   FKACTIONS.  55 


17.   Reduce  to  its  simplest  form  -  of  ^-  of  -^  -*- 1^ 


18.  Divide  ^  of  ^  of  13f  by      ^    . 

19.  Divide  ?i  by  i±i. 

21.Divide|xA,yg^X^^^. 
22.   Simplify  (i  +  ^)x^^?^. 


23.   Simplify  2}  x^^:^^j^ 


9* 


24.    Simplify  54  of  r-^— -  -^  ^2L±_^. 
^    -y    ^       4  +  2^     4i+3f 


To  Find  a  Number  when  a  Fractional  Part  of  it 
IS  Known. 

40.  I.   5f  is  J  of  what  number  ? 

5      3  I  of  some  number  means  the  same  as  1 

f-5_^7_?^     £_16_«i       times  some  number.    If  5|  is  |  times  a  cer- 
'9^72        ^       tain  number,  the  whole  number  is  as  much 
*  as  I  is  contained  in  5|,  which  equals  7^. 

II.   A  boy  after  spending  |  of  his  money  has  $7^  left ; 
how  much  had  he  at  first  ? 

5  If   he  spent  |,  he  had   |  remaining. 

tri^^  —  lEx-  =  —  =  12}f  ^^"^^  t^^  tot^l  sum  is  as  much  as  f  is 

"62^2  "'  contained  in  $7 J,  which  is  $12  J. 

Ans.  $12J. 


5ti  ARITHMETIC. 

EXAMPLES. 

1.  19^  is  ■§■  of  what  number  ? 

2.  32|  is  3§j  of  what  number  ? 

3.  1^  of  II  is  ^^g.  of  what  number  ? 

4.  Y^:f  of  ^  of  ly\  is  f  of  what  number  ? 

5.  ^  of  5y\-  is  6J  times  what  number  ? 

6.  Of  what  number  is  f  the  -J  part  ? 

7.  From  Boston  to  Worcester  is  44  miles,  which  is  f  of 
the  distance  from  Boston  to  Springfield ;  find  the  distance 
from  Boston  to  Springfield. 

8.  Find  the  cost  of  a  barrel  of  flour  when  Jg-  of  a  barrel 
costs  $3.50. 

9.  A  man  can  dig  i|-  of  a  ditch  in  2|  days ;  how  long 
will  it  take  him  to  dig  the  whole  ditch  ? 

10.  A  man  after  selling  ^  of  his  farm  has  28|  acres 
remaining ;  how  many  acres  were  there  in  the  entire  farm  ? 

11.  ^  of  a  basket  of  eggs  were  broken,  and  there  were 
66  left ;  find  the  original  number  in  the  basket. 

12.  A  grocer  sold  f  of  a  barrel  of  sugar  to  one  customer 
and  ^  of  it  to  another  customer,  and  had  45  pounds  left ; 
how  many  pounds  were  there  in  the  barrel  when  full  ? 

13.  In  an  orchard  J  of  the  trees  bear  apples,  J  bear  pears, 
^  bear  peaches,  and  the  remainder,  39  in  number,  bear  plums ; 
find  the  number  of  trees  in  the  orchard. 

14;  A  boy  after  losing  \  of  his  money  has  10  cents  given 
him,  and  then  finds  that  he  has  f  of  the  original  amount ; 
what  was  the  original  amount.  ? 


COMMON  FRACTIONS.  57 


To  Find  what  Fractional  Part  one  Number  is  of 
Another. 

41.   I.   8  is  what  part  of  13  ? 

r^  1  is  y^j  of  13,  and  8  is  8  times  j^^  of  13,  which  is  y\.    The 

— •         number  of  which  a  part  is  taken  is  the  denominator,  and  the 
part  taken  is  the  numerator. 

II.   If  is  what  part  of  3|  ? 

^  5       ^  o        o        q^  We  first  form  a  complex  fraction  exprcss- 

^  =  —  X  ~  =  ^'      ing  the  fractional  part,  and  then  reduce  this 
^  complex  fraction  to  a  simple  one. 


EXAMPLES. 

What  part  of 

1.   12  is  7? 

5. 

15|  is  i? 

9. 

7Jis3i? 

2.   lOi  is  3  ? 

6. 

ttisi? 

10. 

25Jis2|? 

3.   17is4i? 

7. 

*ist? 

11. 

4  is  1  of  6? 

4.   12isf? 

8. 

2|isli? 

12. 

H-l' 

13.  A  of  ^  of  12  is  T^  of  2^? 

14.  l  +  A  +  iis^-F 

15.  12-7f  is  li  +  2f  ? 

16.  37f  is  2ix3f  +  iof  5i? 

17-  i  +  i  +  i  +  iisi-i  +  i-i? 

18-  (J-i)x(4-3f)  is  (2+1)^(3  +  1)? 

19.  If  a  tank  can  be  filled  by  a  pipe  in  11  hours,  what 
part  can  be  filled  in  3^  hours  ? 

20.  If  a  man  can  build  a  wall  in  3J  days,  what  part  can 
he  build  in  2^  days  ? 


58  ARITHMETIC. 

21.  A  man  owning  f  of  a  ship's  cargo  sells  f  of  the  cargo ; 
what  part  of  his  share  does  he  sell  ? 

22.  A  boy  had  $15^  given  him,  and  he  spent  $8 ;  what 
part  of  the  money  did  he  spend  ? 


Reduction  of  Common  Fractions  to  Decimal 
Fractions. 

42.   I.   Reduce  J  to  a  decimal  fraction. 

8)7.000  Since  |  equals  7-^-8,  we  can  perform  the  division 

0.875  decimally  and  obtain  a  decimal  fraction  for  the  value. 

If,  when  the  fraction  is  in  its  lowest  terms,  the  denomi- 
nator contains  any  factor  besides  2  and  5,  the  quotient 
cannot  be  obtained  exactly.  In  such  cases,  as  in  division 
of  decimals,  five  decimal  places  are  ordinarily  enough  for 
the  answer. 

EXAMPLES. 

Reduce  the  following  common  fractions  to  decimal  frac- 
tions : 

1.  f.  4.  ^.  7.  A.  10.  m- 

2.  i.  5.   ^.  a   U.  11.   15^. 

3.  A.  6.  3U.   '         9.  10,%.         12.   62iKt- 

13.  Reduce  -^  to  a  decimal  fraction. 

T 

14.  Express  ^  decimally  to  three  places. 

15.  Reduce  to  decimals  and  add  |,  |-J,  and  9|-J. 

16.  Write  1-^  and  2^  in  decimal  form.  Qive  the  divis- 
ion in  decimals  of  the  first  by  the  second. 


COMMON  FRACTIONS.  59 


Reduction  of  Decimal  Fractions  to  Common 
Fractions. 

43.   I.   Reduce  0.0375  to  a  common  fraction. 

n  nw^  —  _a7  s    _   a  0.0375  can  be  expressed  as  a  common 

—  Toxriyu  ~  'SU-     fraction  in  the  form  x^J^^y.     This  common 
fraction  reduced  to  its  lowest  terms  equals  /j. 

The  denominator  of  the  common  fraction  is  always  1 
with  as  many  zeros  annexed  as  there  are  decimal  places  in 
the  decimal  fraction. 


EXAMPLES. 

Reduce  the  following  decimal  fractions  to  common  frac- 
tions : 


1. 

0.7. 

7. 

0.0625. 

13. 

0.00096. 

2. 

0.24. 

8. 

0.0806. 

14. 

21.1875. 

3. 

0.625. 

9. 

0.98. 

15. 

0.05128. 

4. 

0.440. 

10. 

12.043. 

16. 

14.06225. 

5. 

0.0016. 

11. 

0.03125. 

17. 

42.030125. 

6. 

5.082. 

12. 

8.65. 

18. 

0.0007648267. 

Reduction  of  Common  Fractions  to  Circulating 
Decimals. 

44.  When  the  result  cannot  be  obtained  exactly  in  re- 
ducing a  common  fraction  to  a  decimal,  if  the  division  be 
carried  far  enough,  the  quotient  will  be  found  to  contain 
the  repetition  of  a  figure  or  series  of  figures.  For  example, 
2  =  0.6666  +  ;  2V  =  0.3181818  -f .    Such  decimals  are  known 


60  ARITHMETIC. 

as  circulating  decimals,  repeating  decimals,  or  infinite  deci- 
mals. The  figure  or  series  of  figures  which  is  repeated  is 
called  the  repetend.  In  the  case  of  a  single  figure  the  repe- 
tend  is  denoted  by  a  dot  over  the  figure,  and  in  the  case  of 
a  series  of  figures  by  dots  over  the  first  and  last  figures. 
For  example,  |  =  0.6 ;  ^^  =  O.SiS. 

I.  Reduce  ^  to  a  circulating  decimal. 

0.3428571 
,35)12.0 
105 


150 
140 
100 
70 


300 

280 


We  must  continue  the  division  until  the  re- 
mainder is  the  same  as  some  preceding  remainder ; 
from  this  point  the  figures  will  continue  in  series 
as  before.  In  the  present  example  15  is  the  same 
as  the  second  remainder.  The  repetend  begins 
with  4  and  ends  with  1 ;    hence  dots  are  placed 


200 

-^'^  over  these  figures. 

250 
245 


50 
35 
15 


EXAMPLES. 


Reduce  the  following  common  fractions  to  circulating 
decimals : 

1.  i.  5.   f  9.   SA-  13.   2^. 

2.  A-  6.   Jj.  10.  ,1^.  14.   ^. 

3.  f  7.   S^.  11.   ^j.  15.   12^. 

4.  5Jj.  8.   n.  12.   18^.  16.   5^1^. 

17.    What  circulating  decimal  is  equivalent  to  the  sum  of 

1     1     oTirl     1    9 

3,  y,  ana  yy  .'■ 


COMMON   FRACTIONS.  61 

18.  Find  the  sum  of  6J,  1^,  and  8^%,  and  express  the 
result  as  a  circulating  decimal. 

19.  Reduce    ^  ^TF  +  it)  to  a  repeating  decimal. 


Reduction  of  Circulating  Decimals  to  Common 
Fractions. 

45.  When  the  repetend  comprises  all  the  decimal  places, 
a  circulating  decimal  is  equal  to  a  common  fraction  which 
has  the  repetend  for  the  numerator  and  as  many  nines  as 
there  are  decimal  places  for  the  denominator.  Take  0.324 
as  an  example  to  show  this. 

1000  times  0.324  =  324.324324+. 
Once  0.324  =     0.324324  +. 
By  subtraction  we  obtain 

999  times  0.324  =  324. 
Hence  0.324  =  ff|. 

I.  Reduce  0.72  to  a  common  fraction. 

n  79  _  72  _   8  '^^^  denominator  is  99.     Then  ^f  must  be 

~"9T  ~~  TT-      reduced  to  its  lowest  terms,  which  is  ^. 

II.  Reduce  0.4772  to  a  common  fraction. 
0.4772  =  0.47||  =  0.47A  =  ^  =  ^  =  |. 

The  repetend  reduces  to  the  common  fraction  j*y.  Then  the  decimal 
0.47 y\  can  be  expressed  as  a  complex  fraction,  which  reduces  to  \\- 

When  circulating  decimals  are  to  be  added,  subtracted, 
multiplied,  or  divided,  they  should  first  be  reduced  to  com- 
mon fractions ;  then  perform  the  operations  indicated,  and 
reduce  the  resulting  fraction  to  a  decimaL 


62 


ARITHMETIC. 


EXAMPLES. 

Keduce  the  following  circulating  decimals  to  common 
fractions : 


1.  0.3. 

2.  0.27. 

3.  0.0027. 

4.  0.0127. 

5.  0.216. 

6.  7.0136. 

7.  0.20054. 


8.  2.00054. 

9.  4.608i. 

10.  0.4081. 

11.  15.i08. 

12.  0.225. 

13.  0.00225. 

14.  0.857142. 


15.  3.2343. 

16.  0.002343. 

17.  0.012343. 

18.  10.002343. 

19.  12.03405. 

20.  0.81247. 

21.  1.15479li. 


22.  What  common  fraction  equals  the  sum  of  0.18  and 
0.307692  ? 

23.  Add  0.03  to  0.462,  expressing  the  result  as  a  circu- 
lating decimal. 

24.  Add  0.07  to  0.382,  expressing  the  result  as  a  circu- 
lating decimal. 

25.  Multiply  0.145  by  0.297,  and  give  the  answer  as  a 
circulating  decimal. 

26.  Multiply  0.3461538  by  0.285714,   and  express   the 
result  as  a  circulating  decimal. 

27.  Multiply  2.604   by  1.234,  and  divide  the  result  by 
0.004. 

Greatest  Common  Divisor  oe  Fractions. 


46.   I.    Find  the  G.C.D.  -of  ^,  ^,  and  ||. 
G.C.D.  of    3,    9,  and  12  =      3. 
L.C.M.  of  20,  10,  and  25  =  100. 


G.C.D. 


3 


The  G.C.D.  of  the  numera- 
tors is  3.  To  be  a  divisor  of 
7^^  this  number  must  be  divided 
bj  20;  to  be  a  divisor  of  /^  it 


COMMON  FRACTIONS.  63 

must  be  divided  by  10 ;  and  to  be  a  divisor  of  ^f  it  must  be  divided  by 
25,  If,  however,  3  be  divided  by  the  L.C.M.  of  the  denominators,  it 
will  be  divided  by  the  least  number  containing  air  the  factors  of  the 
denominators ;  hence  this  result  is  the  G.C.D.  of  the  fractions. 

To  find  the  G.C.D.  of  several  fractions,  write  the  G.C.D. 
of  the  numerators  over  the  L.  CM.  of  the  denominators.  The 
fractions  should  be  in  their  lowest  terms,  and  mixed  num- 
bers should  first  be  reduced  to  improper  fractions. 

EXAMPLES. 
Find  the  G.C.D.  of 

1.  t,i,andf  5.  3i,2i,andf 

2.  f,i|,andf^.  6.  6|,  IGf,  and  6f 

3.  ^,}i,and|^.  7.  GJ,  84,  and  12J. 

4.  1  \,  i,  and  4.  8.  1  A,  h\.  and  1^.- 

9.  Find  the  width  of  the  widest  stone  that  can  be  used 
in  laying  three  walks  which  are  respectively  3}  feet,  3^ 
feet,  and  54  feet  wide. 

10.  What  is  the  largest  measure  that  can  be  used  in 
measuring  the  contents  of  four  bins  which  contain  respec- 
tively 9,  134,  10|,  and  lOJ-  bushels  ? 

Least  Common  Multiple  of  Fractions. 

47.   I.   Find  the  L.C.M.  of  /^,  3%,  and  4|. 

L.C.M.  of    3,     9,  and  12  =  36.  The  L.C.M.  of  the  numera- 

G.C.D.  of  20,  10,  and  25  =    5.        tors  is  36,  which  is  also  a  mul- 

J  p-vT  36—71  *^P^®  °^  *^^  fractions.     If  this 

^  ^'  number  be  divided  by  20  or  any 

factor  of  20,  it  is  still  a  multiple  of  2% ;  if  divided  by  10  or  any  factor 
of  10,  it  is  still  a  multiple  of  ^-^ ;  if  divided  by  25  or  any  factor  of  25, 
it  is  still  a  multiple  of  if.     If,  then,  36  be  divided  by  the  G.C.D.  of  20, 


64  ARITHMETIC. 

10,  and  25,  it  will  be  divided  by  the  greatest  common  factor  of  the 
denominators  ;  hence  this  result  is  the  L.C.M.  of  the  fractions. 

To  find  the  L.C.M.  of  several  fractions,  write  the  L.C.M. 
of  the  numerators  over  the  G.C.D.  of  the  denominators. 

EXAMPLES. 
Find  the  L.C.M.  of 

1.  f,T\,  and^.  5.  2f,  31,  and  43^. 

2.  -I,  i,  and  -|.  6.  Ill  142,  and  33^. 

3.  I,  i,  I,  and -I-.  7.  4^,  5^^,  and  43^- 

4.  ^,  ^V.  and  A-  8.  ^\,  ^,  2|,  5,  and  6^. 

9.  Find  the  capacity  of  the  smallest  tank  whose  contents 
can  be  exactly  measured  by  either  of  three  measures  which 
contain  respectively  1^,  If,  and  2^  quarts. 

10.  A  can  travel  around  a  certain  island  in  2^  days,  B 
in  3^  days,  and  C  in  3J  days.  If  they  set  out  at  the  same 
time  from  the  same  point,  and  travel  in  the  same  direction, 
in  how  many  days  will  they  all  come  together  at  the  start- 
ing point,  and  how  many  times  will  each  man  have  gone 
around  the  island  ? 

11.  The  pendulum  of  one  clock  makes  24  beats  in  26  sec- 
onds ;  that  of  another,  36  beats  in  40  seconds.  If  they  start 
at  the  same  time,  when  first  will  the  beats  occur  together  ? 

MISCELLANEOUS    EXAMPLES. 

1.  Add  |i,  I,  and  A  of  f .        3.   Add  A  to  ?i^. 

2.  Add  i,  I  of  I,  and  If-  4.    Add  ^f  of  -i"  *«  '^' 

5.  Add  f  of  3i  to  4  of  i?iof  -A. 

loj 


COMMON  FRACTIONS.  65 

G.   Add  f  of  18  j3_  and  |i  of  |  of  G^. 

7.  Add  i  of  f  of  28||  to  8^. 

8.  Add  |i,  I,  and  I  of  A. 

9.  Add^itofof  ifof  |of  (J-.+). 


6J 


10.   Add  J  to 
o 


^*T)' 


'5i 


11.   Add  ?  of  '^  to  I  of  (4i  -  2J). 

"8" 

.o     All  ^-^007       .    „ 

''•  ^^'^'  ao3^  ""^'  *• 

13.  Add  l-LM  and  ?i  +  ?i. 

i  +  4i  7i-4| 

14.  What  is  the  sum  of  t^  and  i^LA  ? 

i  J  of  2i 

.  P,     .  .  1   1  of  2i  .    0.06  +  0.3^ 

16.  Subtract  J  of  ^-^  from  ^^j^- 

17.  From  }  of  f  take  ^  of  |. 


4i 


18.  From  f  of  ^  take  ^  of  1|. 

19.  From  ^T^  ^  ^^  subtract  ^. 

5i-4J  2i 

4.  Ql    ,02 

20.  Find  the  difference  between  -3-  and  -^ 2. 

If  7 


^±1, 


21.    Find  the  difference  between  3J  x  6||  and  ^ 


66  ARITHMETIC. 

22.  Subtract  7-i-  -f  A  of  tt  ^^^^^  15^  +  -|-  +  OM. 

23.  From  34  subtract  (—  of  ii  of  l*^  h-  t^- 

24.  From  54  subtract  ?i  ^  /^A  of  ^  of  4iY 

^  3i       VlO       2|         V 

25.  Find  the  sum  and  product  of  -J,  ^,  and  f . 


26.   Divide  |  of  7|  by  f  of  12U. 

3 
4* 


27.  Divide  (±-1)  by 

28.  Divide  A  _  1  by  A 


6^     7     -^   11 


29.    Divide  it  by  —  of  (^-l\ 


30. 

Diviflp  14  of    9    nf  12   hv             "^ 

TOt,eOtlTby^_^^^^ 

31. 

Divide.^|-l|by|-of(^^  +  |). 

32. 

Divide  0.75  by  ?i  X  0.081. 
lo 

33.  Reduce  to  a  common  denominator  and  add  f  X  |  X  |, 
A.  i,  and  j%. 

34.  Find  the  simplest  expression  for 1:  +  _j:  _  JI_. 

^  ^  3^      9       2      44. 

35.  Add  ^,  y\,  -^,  and  ^,  and  reduce  the  sum  to  a  deci- 
mal fraction  carried  to  three  decimal  places. 

36.  Add  f ,  2|-,  f,  and  \^,  and  divide  the  sum  by  fifty-six 
thousandths. 

37.  From  ^\  of  1|  subtract  ^  of  ^,  and   reduce  the 
answer  to  a  decimal. 


COMMON  FRACTIONS.  67 

38.  From  }  of  |-|  subtract   ^  of  2J,  and  reduce  the 
answer  to  a  decimal. 

39.  Divide  (2|  x  j\)  by  (2J  - 1^),  and  reduce  the  result 
to  a  decimal. 

40.  Divide  J  of  ^  of  f  by  j^,  and  add  the  quotient  to  -f. 

41.  Divide  f  of  ^^  of  |-  by  ~^,  and  add  the  quotient  to 


5A' 


f-A- 

42.  Divide  (—  —  --\-~\  by  f ,  and  reduce  the  resnlt  to 
an  equivalent  decimal  fniction. 

43.  From  \  of  1^  take  ^,  add  to  the  remainder  |,  and 

divide  the  result  by  6f .  2" 

on 

44.  From  the  sum  of  --^  and  —  subtract  44,  and  divide 

13J  1  ^' 

the  result  by  the  product  of  3J  and  2^. 

45.  To  f  of  f  add  y\-  -^  ^,  multiply  the  sum  by ,  and 

2.  of    6  ^^ 

divide  the  product  by  ^— — ^• 

46.  Add  -^  and  ^;  divide  the  result  by  7^,  and 
change  the  quotient  to  a  decimal. 

47.  From  ^  of  2f  subtract  the  product  of  0.075  and  1^ 
and  divide  the  remainder  by  12;  reduce  the  result  to  a 
decimal  form. 

31. 

48.  From  f  of  f  subtract  -|  of  —2-,  add  to  the  remainder 

^,  divide  the  result  by  6J,  and  change  the  quotient  to  a 
decimal. 


68  ARITHMETIC. 

49.  Eeduce     ^^  of  4^  of  f  to  a  simple  fraction. 

5  +  t 

50.  Reduce    ^  ^     "^    to  a  decimal  fraction. 

31  + 1.125 

2      ^  —  ^  X  — - 

51.  Express  as  a  decimal  ^  x  ^      '^       ^  • 

52.  What  decimal  is  equivalent  to  |  of  ^  X  0.021  ? 

53.  Simplify  (l  +  i±i)  ^  (l  +  ^). 

54.  M4.5:5?_Zi=what? 
3-V^  33       27 

55.  (1 J  +  H  -  ^-^24)  -  (151  - 1.209)  =  what  ? 

02 

29  12 

56.  Reduce ; to  its  simplest  decimal  form. 

1300      41.64 

57.  Snnphfy3j^X^-+3|j. 

58.  Reduce  ^^  ^  1^^  X  ^  X  ^  to  a  decimal. 

V41-     2y      5     2 

59.  Simplify L^ 

2  +  - 


4H 


«+i 


60.   What  is  the  exact  value  of  ('2|+'f  of  ^  +  i")  .^  i^-. 


T? 


~      ~  ~  i  nf  i     "■" 


61.   Simplify  ^^ZJE  of  ^~^  of  1—1  of  585. 


COMMON   FRACTIONS.  69 

62.  The  sum  of  |  and  ^^  is  diminished  by  yi^.  How  many 
times  does  the  difference  contain  ^  of  the  sum  of  J,  ^, 
and  jV  '^ 

63.  The  sum  of  two  numbers  equals  3^,  and  one  of  them 

is  the  difference  between  — '^  and  — ^ ;  what  is  the  other 

11  9 

number  ? 

64.  -J-  of  a  number  exceeds  y^j^  of  it  by  15 ;  what  is  the 
number  ? 

G5.  What  part  of  ^  is  i^? 
^  i 

m.   What  part  of  24  is  ^  x  ^^~^^  ? 
^  ^      31|  "^  ^  X  3f 

67.  Simplify  ^j  -  0.042  -  2.4  +  Tj 

16  J^  _j-  60^ 

68.  Simplify  (3.71-1.9Q8)x7.03, 
2.8  of  2.27 


69.    Simplify 


1.136 


70.  Simplify  ^±^^  +  _ii_  «  A  ^  1. 

13^         Q,    1     12     17 

71.  Simplify  (2^of  3yV)+t-(liof  l,^)-(l|of  4^of  ,^> 

72.  Find  the  G.C.D.  and  the  L.C.M.  of  J,  1|^,  and  3.60. 

73.  The  sum  of  ^  ^'^•^H  and  ?i^  is  how  many  times 

0.5  31 

their  difference  ? 

74.  The  sum  of  M-^^  and  i^^i  is  how  many  times 
.^  .     -..^  «        0.5  I X  2.25  -^ 
their  difference  ? 


70  ARITHMETIC. 

75.  What  is  the  value  of  (^-^^  +  i^J^  -  ^  -7^^  ? 

rra     o-       r*         54-!-f      ^  2     .   U  of  41        1,2 

76.  Simpnfy 2^ — ^ —  X  -  of  —^ ^ 

77.  Simplify  ^,^^^+ffi~_ff°  and  3  X  (3|  x  5f )  x  17f , 
and  find  their  sum. 

78.  Find  the  vahie  of  (^4|  --  ^i±i^  x  0.3G  x  0.236. 

79.  Simplify 

0.6  of  3.3  -f  ^—  of  17  +  0.4  of  5.75  -  ^•^^^^^^. 
^.625  2.095238 


80.  By  what  must  ^  be  multiplied  to  give  the  product  1? 

81.  What  number  is  that,  -f  of  which  exceeds  \  of  it 
by  Hi? 

82.  Find  the  cost  of  81|  acres  of  land  at  f  28|-  per  acre. 

83.  Find  the  weight  of  8^  reams  of  paper  at  14-f^  pounds 
per  ream. 

84.  Find  the  cost  of  |^  of  a  ton  of  coal  if  3  tons  cost  $20. 

85.  If  a  man  saw  3 J  cords  of  wood  in  one  day,  how  much 
will  he  saw  in  |  of  a  day  ? 

86.  Find  the  price  of  flour  per  barrel  when  9f  barrelf 
cost  $65f . 

87.  At  %2\  per  barrel,  how  many  barrels  of  apples  can 
be  bought  for  f  55  ? 

88.  If  a  man  travel  28-f^  miles  in  one  day,  how  many  days 
will  it  take  him  to  travel  177|  miles  ? 


COMMON   FRACTIONS.  71 

89.  If  a  man  walk  3J  miles  in  |  of  an  hour,  at  what  rate 
does  he  walk  per  hour  ? 

90.  What  number  divided  by  ^  equals  6^  ? 

91.  Find  the  cost  of  8  rolls  of  carpet,  42\  yards  in  a  roll, 
at  91J  cents  a  yard. 

92.  If  f  of  a  yard  of  cloth  cost  f  3^,  what  is  the  cost  of 
4|  yards  ? 

93.  A  farmer  sold  4|^  tons  of  hay  at  the  rate  of  2|  tons 
for  f  44 ;  what  did  he  receive  for  it  ? 

94.  rind  the  number  of  square  yards  in  the  surface  of 
three  floors  measuring  respectively  16J,  21-}-|,  and  28^ 
square  yards. 

95.  A  farm  is  divided  into  four  fields  which  contain  re- 
spectively 18 1,  22|^,  19i^j  and  29^^  acres;  find  the  number 
of  acres  in  the  entire  farm. 

96.  What  number  is  that,  to  which  if  you  add  J  of  19|, 
the  sum  will  be  150  ? 

97.  A  man  bought  95  bushels  of  corn  at  33J  cents  a 
bushel  and  sold  it  at  37^  cents  a  bushel ;  find  the  amount 
gained. 

98.  What  number  is  that,  ^  of  which  exceeds  2\  by  13|  ? 

99.  A  merchant  sold  38  yards  of  cloth  at  the  rate  of  2^ 
yards  for  $3 ;  what  did  he  receive  for  it  ? 

100.  What  is  the  price  of  land  per  acre  when  ^  of  an 
acre  costs  $44.25  ? 

101.  The  product  of  three  numbers  is  453 J;  two  of  them 
are  5f  and  11 J ;  find  the  third  number. 

102.  If  J  of  a  ton  of  hay  will  pay  for  8  barrels  of  apples 
worth  $2 J  per  barrel,  what  is  the  value  of  the  hay  per  ton  ? 


72  ARITHMETIC. 

103.  If  j^  of  a  yard  of  velvet  cost  f  T-f,  how  many  yards 
can  be  bought  for  $68if  ? 

104.  A  clerk  spends  $425  a  year  for  board,  which  is  ^J 
of  his  salary ;  what  is  his  salary  ? 

105.  If  5^  tubs  of  butter  cost  $103|,  how  many  bar- 
rels of  flour  worth  f  8J  per  barrel  will  pay  for  one  tub  of 
butter? 

106.  If  I  of  I  of  a  ship  cost  $70000,  what  is  -^  of  it 
worth  ? 

107.  How  many  pieces  of  cloth,  each  containing  2-^  yards, 
can  be  cut  from  a  piece  50^  yards  in  length  ? 

108.  rind  the  cost  of  8J  tons  of  hay  when  2\  tons  cost 
$31J. 

109.  A  farmer  exchanged  10  pounds  of  butter  worth  31J 
cents  a  pound  for  sugar  worth  7|-  cents  a  pound ;  how  much 
sugar  did  he  obtain  ? 

110.  If  a  certain  number  is  increased  by  |-  of  i|-  of  itself, 
the  result  is  246 ;  find  the  number. 

111.  A  boy  spent  -^  of  his  money  one  day,  and  f  of  it 
the  next  day,  and  then  had  65  cents  left  j  how  much  had  he 
at  first  ? 

112.  A,  owning  -J  of  a  farm,  sold  J  of  his  share  to  B,  and 
-J-  of  what  he  then  owned  to  C  for  $420 ;  what  was  the  value 
of  the  entire  farm  at  the  same  rate  ? 

113.  A  tailor  has  97^  yards  of  cloth,  from  which  he 
wishes  to  cut  an  equal  number  of  coats  and  vests;  how 
many  of  each  can  he  cut  if  they  contain  4J  and  1|-  yards 
respectively  ? 

114.  If  I  of  a  ton  of  coal  cost  $6},  what  will  -f^  of  a 
ton  cost? 


COMMON  FRACTIONS.  73 

115.  A  horse  and  cow  were  bought  for  ^180,  and*  the  cow 
cost  -J  as  much  as  the  horse ;  find  the  price  of  each. 

116.  If  I  of  a  bushel  of  corn  be  worth  ^  of  a  bushel  of 
wheat,  and  wheat  be  worth  $1.40  a  bushel,  how  many  bushels 
of  corn  can  be  bought  for  $27  ? 

117.  ^  of  ^  of  28  times  what  number  equals  50|? 

118.  A  man,  owning  |-  of  a  mill,  sold  -^  of  his  share  for 
$2750 ;  find  the  value  of  the  whole  mill  at  the  same  rate. 

119.  A  man  has  ^  of  his  property  invested  in  real  estate, 
J  in  state  bonds,  ^  in  bank  stock,  and  the  remainder,  $5500, 
in  business ;  find  the  value  of  his  entire  property. 

120.  A  owns  ^  of  a  mill,  and  B  the  remainder;  |  of  the 
difference  between  their  shares  is  $10500;  find  the  value 
of  the  whole  mill. 

121.  A  farmer  sold  21^  dozen  eggs  at  18|  cents  a  dozen, 
and  bought  14J  yards  of  cloth  at  12^  cents  a  yard;  how 
much  money  did  he  have  left  ? 

122.  If  19  pounds  of  butter  cost  $6.33^,  what  part  of  a 
pound  can  be  bought  for  25  cents  ? 

123.  What  is  the  smallest  sum  of  money  that  can.be 
exactly  paid  either  in  pieces  of  money  worth  6^  cents  or  in 
pieces  worth  8J  cents  ? 

124.  Find  the  width  of  the  widest  blocks  that  will  exactly 
fit  either  of  three  walks  which  are  respectively  6J,  7^,  and 
10  feet  wide. 

125.  If  6  be  added  to  both  terms  of  the  fraction  ^,  is 
the  value  of  the  fraction  increased  or  diminished,  and  how 
much  ? 

126.  If  6  be  subtracted  from  both  terms  of  the  fraction 
■^,  is  the  value  of  the  fraction  increased  or  diminished,  and 
how  much  ? 


74  ARITHMETIC. 

127.  If  6  men  can  do  a  piece  of  work  in  |  of  f  of  ^  of 
6J  days,  how  many  men  could  do  it  in  one  day  ? 

128.  $48|  are  to  be  divided  among  5  men  and  3  boys  so 
that  each  boy  will  have  half  as  much  as  a  man ;  how  much 
will  each  have  ? 

129.  A  grocer  sold  f  of  a  barrel  of  flour  to  one  customer, 
|-  of  the  remainder  to  another  customer,  and  had  24-|-  pounds 
left;  how  many  pounds  were  there  in  the  barrel  when  full? 

130.  A  merchant  owned  ^  of  a  stock  of  goods ;  -f-  of  the 
whole  stock  was  destroyed  by  fire,  and  -^  of  the  remainder 
damaged  by  water.  How  much  did  the  merchant  lose,  pro- 
vided the  uninjured  goods  were  sold  at  cost  for  $4200,  and 
the  damaged  at  half  the  cost  ? 


COMPOUND   NUMBERS.  76 


CHAPTER  V. 
COMPOUND  NUMBERS. 

48.  When  the  value  of  anything  is  expressed  in  different 
units  of  the  same  nature,  it  is  called  a  compound  number ; 

as  3  bushels  2  pecks  5  quarts. 

49.  Long  or  Linear  Measure  is  used  in  measuring  lengths 
and  distances. 

TABLE. 

12  inches  (in.)  =  1  foot  (ft.). 

3  feet  =lyard  (yd.). 

5 J  yards  or  16|-  feet  =  1  rod  (rd.). 
320  rods  or  5280  feet  =  1  mile  (mi.). 

Note.   A  line  =  ^^  in. ;  a  furlong  =  40  rd. ;  a  fathom  =  6  ft. 

50.  Surveyors'  Measure  is  used  in  measuring  dimensions 

of  land. 

TABLE. 

7.92  inches  =  1  link  (li.). 
100  links     =  1  chain  (ch.). 
80  chains  =  1  mile  (mi.). 

Note.   A  surveyors'  chain  is  4  rods  long  and  contains  100  links. 
Engineers  use  a  chain,  or  measuring  tape,  100  feet  long. 

51.  Square  Measure  is  used  in  measuring  the  area  of 
surfaces. 


76  ARITHMETIC. 


TABLE. 


144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.). 

9  square  feet  =  1  square  yard  (sq.  yd.). 

301  square  yards  or  7  ^  j  /         j  \ 

0T01  -p    4.        C  =  ^  square  rod  (sq.  rd.). 

272J  square  feet       ) 

160  square  rods  =  1  acre  (A.). 

640  acres  =1  square  mile  (sq.  mi.). 

Note.  A  perch  (P.)  is  a  square  rod,  and  a  rood  (R.)  =  40  sq.  rd. 

10  square  chains  =  1  acre. 

A  section  of  land  is  a  square  mile ;  36  sections  =  1  township. 

52.  Cubic  Measure  is  used  in  measuring  things  which  haVe 
length,  breadth,  and  thickness. 

TABLE. 

1728  cubic  inches  (cu.  in.)  =1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.). 

53.  "Wood  Measure  is  used  in  measuring  wood  and  other 
merchandise. 

TABLE. 

16  cubic  feet  =  1  cord  foot  (cd.  ft.). 

8  cord  feet  or  128  cubic  feet  =  1  cord  (cd.). 

Note.  A  cord  of  wood,  as  generally  piled,  is  8  ft.  long,  4  ft.  wide, 
and  4  ft.  high. 

54.  Liquid  Measure  is  used  in  measuring  liquids. 

TABLE. 

4  gills  (gi.)  =lpint  (pt.). 
2  pints  =1  quart  (qt.). 

4  quarts        =1  gallon  (gal.). 

Note.  A  gallon  contains  231  cu.  in.  31 1  gallons  are  considered  a 
barrel  (bbl.),  and  63  gallons  a  hogshead  (hhd.)  ;  but  barrels  and  hogs- 
heads are  made  of  various  sizes. 


COMPOUND  NUMBERS.  T7 

55.  Apt/cliecaries'  Fluid  Measure  is  used  in  compounding 
medicines. 

TABLE. 
60  minims  {%)  =1  fluid  dram  (f  3). 

8  fluid  drams     =  1  fluid  ounce  (f  S  ). 
16  fluid  ounces    =  1  pint  (0.). 

56.  Dry  Measure  is  used  in  measuring  dry  articles. 

TABLE. 
?  pints  (pt.)  =1  quart  (qt.). 
8  quarts         =  1  peck  (pk.). 
4  pecks  =  1  bushel  (bu.). 

Note,   a  oushel  contains  2160.42  cu.  in. 

57.  Troy  Weight  is  used  in  weighing  gold,  silver,  and 
precious  stones. 

TABLE. 

24  grains  (gr.)     =1  pennyweight  (pwt.). 
20  pennyweights  =  1  ounce  (1  oz.). 
12  ounces  =  1  pound  (lb.). 

Note.  1  lb.  Troy  =  5760  grains.  In  weighing  diamonds  1  carat  = 
3^  Troy  grains,  and  is  divided  into  quarters,  which  are  called  carat 
grains. 

The  word  carat  applied  to  gold  indicates  the  number  of  parts  in  24 
that  are  pure  gold.  For  example,  18  carats  fine  means  that  ||  is  pure 
gold,  while  the  rest  is  alloy. 

58.  Apothecaries'  Weight  is  used  in  compounding  medi- 
cines 

TABLE. 

20  grains  (gr.)  =  1  scruple  (3). 
3  scruples  =  1  dram  (3). 

8  drams  =  1  ounce  (  5  )  • 

12  ounces  =1  pound  (lb.). 

Note.  The  pound,  ounce,  and  grain  have  the  same  weight  as  those 
of  Troy  Weight. 


78  ARITHMETIC. 

59.  Avoirdupois  Weight  is  used  in  weighing  all  articles 
except  gold,  silver,  and  precious  stones. 

TABLE. 

16  drams  (dr.)  =1  ounce  (oz.). 

16  ounces  =  1  pound  (lb.). 

100  pounds  =1  hundred-weight  (cwt.). 

20  hundred-weight  or  )  _  ..  ,       .rp  ^ 

2000  pounds  )           on  (^   .;. 

Note.   1  lb.  Avoirdupois  —  7000  gr. 

The  long  ton  of  2240  lb.,  and  the  long  hundred-weight  of  112  lb.,  are 
used  at  United  States  Custom  Houses  and  in  wholesale  transactions  in 
coal  and  iron.     The  ton  of  2000  lb.  is  often  called  the  short  ton. 

1  quarter  (qr.)  =  25  lb. ;  when  the  long  ton  is  the  standard,  1  qr.  =  28  lb. 

60.  Circular  or  Angular  Measure.  A  circle  is  a  plane 
figure  bounded  by  a  curved  line,  every  point  of  which  is 
equally  distant  from  a  point  within  called  the  centre.  The 
bounding  line  is  called  the  circumference,  and  any  part 
of  the  circumference  is  called  an  arc.  A  straight  line  pass- 
ing through  the  centre  and  having  its  extremities  in  the 
circumference  is  called  a  diameter;  a  straight  line  drawn 
from  the  centre  to  the  circumference  is  called  a  radius,  and 
it  is  equal  to  one  half  a  diameter.  The  circumference  is 
divided  into  360  equal  parts,  called  degrees,  each  degree 
into  60  minutes,  and  each  minute  into  60  seconds. 

The  opening  between  two  straight  lines  which  meet  at  a 
point  is  called  an  angle,  and  the  point  where  the  lines  meet 
is  called  the  vertex  of  the  angle.  An  angle  with  its  vertex 
at  the  centre  of  a  circle  is  measured  by  the  arc  included 
between  its  sides.  The  length  of  an  arc  of  one  degree 
varies  with  the  size  of  the  circle,  but  an  angle  of  one 
degree  always  has  the  same  size  opening  between  the  two 
lines. 


COMPOUND   NUMBERS.  79 

The  annexed  diagram  represents  a 
circle ;  G  is  the  centre,  DB  an  arc,  AB 
a   diameter,    and   CD  a  radius.     The 
angle  DCB  contains  the  same  number  ^ 
of  degrees  as  the  arc  DB. 

TABLE. 

60  seconds  (")=  1  minute  ('). 
60  minutes       =  1  degree  (°). 
360  degrees        =  1  circumference. 

Note.  An  arc  of  90°  is  called  a  quadrant,  and  an  angle  of  90°  is 
called  a  right  angle. 

A  degree  of  longitude  at  the  equator,  or  a  degree  of  latitude,  equals 
69.16  miles. 

61.  The  measures  of  time  are  determined  by  the  revolu- 
tion of  the  earth  on  its  axis  and  around  the  sun. 

TABLE. 

60  seconds  (sec.)  =  1  minute  (min.). 

60  minutes  =lhour(hr,). 

24  hours  =  1  day  (da.). 

7  days  =  1  week  (wk.). 

365  days  =lyear(yr.). 

366  days  =  1  leap  year. 
100  years  =  1  century. 

The  length  of  a  solar  day  is  the  interval  between  two 
successive  transits  of  the  sun  over  the  same  meridian.  The 
length  of  a  civil  day  is  the  interval  between  two  successive 
midnights,  and  is  the  average  length  of  all  the  solar  days 
in  the  year. 

The  exact  time  in  which  the  earth  revolves  about  the  sun 
is  365  da.  5  hr.  48  min.  49.7  sec.  For  convenience  in  reck- 
oning it  is  necessary  to  have  an  integral  number  of  days  in 


80  ARITHMETIC. 

a  year,  so  it  lias  been  arranged  to  let  the  common  year  con- 
sist of  365  days,  while  certain  years,  called  leap  years, 
consist  of  366  days.  When  the  number  denoting  the  year  is 
divisible  by  4  and  not  by  100,  or  is  divisible  by  400,  the  year 
is  a  leap  year.  For  example,  1884  and  2000  are  leap  years, 
but  1885  and  1900  are  common  years.  By  this  method  of 
reckoning  the  error  is  less  than  1  day  in  3600  years. 

The  year  is  divided  into  12  months  (mo.).  Their  names 
and  the  number  of  days  in  each  are  given  in  the  following 
table : 

January  (Jan.) 31. 

February  (Feb.)  .     .  28;  in  leap  year  29. 

March  (Mar.) 31. 

April  (Apr.) 30. 

May .  31. 

June 30. 

July 31. 

August  (Aug.)     . 31. 

September  (Sep.  or  Sept.)     .     .     .     .30. 

October  (Oct.) 31. 

November  (Nov.) 30. 

December  (Dec.) 31. 

In  business  it  is  customary  to  reckon  30  days  to  a  month, 
which  makes  an  error  of  5  days  a  year. 

Note.  The  number  of  days  in  each  month  can  easily  be  remembered 
by  the  following  stanza : 

Thirty  days  hath  September, 
April,  June,  and  November ; 
All  the  rest  have  thirty-one, 
Except  February  alone, 
To  which  we  twenty-eight  assign, 
Till  leap  year  gives  it  twenty-nine. 


COMPOUND  NUMBERS.  *81 

62.  English  or  Sterling  Money  is  the  currency  of  Great 

Britain  and  many  of  its  colonies. 

TABLE. 

4  farthings  (far.)  =  1  penny  (d.). 
12  pence  =  1  shilling  (s.). 

20  shillings  =  1  pound  (£). 

Note.  A  florin  =  28.;  a  crown  =  58.;  a  sovereign  =  20  s. ;  a  guinea 
=  21  8. 


63.  Miscellaneous  Tables. 

NUMBERS.  PAPER. 

12  units  =  1  dozen  (doz.).  24  sheets    =  1  quire. 

12  dozen  =  1  gross  (gro.).  20  quires    =  1  ream. 

12  gross  =  1  great  gross.  2  reams     =  1  bundle. 
20  units  =  1  score.  5  bundles,=  1  bale. 

BOOKS. 

in    2  leaves  is  a  folio, 
in    4  leaves  is  a  quarto  or  4to. 
in    8  leaves  is  an  octavo  or  8vo. 
in  12  leaves  is  a  12mo. 
in  16  leaves  is  a  16mo. 
in  18  leaves  is  an  18mo. 
.  in  24  leaves  is  a  24mo. 

Note.  These  names  are  based  on  sheets  measuring  about  18  in.  X 
24  in. 


A  book  formed  of 
sheets  folded 


Reduction  Descending. 

64.  The  process  of  changing  a  compound  number  from 
one  denomination  to  another  without  altering  its  value  is 
called  reduction.  When  the  reduction  is  from  a  higher  to 
a  lower  denomination,  it  is  called  reduction  descending. 


82  ARITHMETIC. 

I.   Reduce  8  lb.  6  oz.  8  pwt.  21  gr.  to  grains. 

8  lb.  6  oz.  8  pwt.  21  gr. 
12 

96 

6  Since  there  are  12  oz.  in  1  lb.,  in 

302  OZ.  ^  lb-  there  are  8  times  12  oz.,  which 

20  equals   96   oz. ;  6  oz.  added  to   this 

oqTq  gives  102  oz.     In  1  oz.  there  are  20 

o  pwt. ;  in  102  oz.  there  are  102  times 

20  pwt.,  or  2040  pwt.;  8  pwt.  added 

gives  2048  pwt.     In  1  pwt.  there  are 

24  gr. ;  in  2048  pwt.  there  are  2048 

times  24   gr.,  or  49162  gr.;  21  gr. 

added  gives  49173  gr. 


2048  pwt. 

24 
8192 
4096 


49152 

21 

49173  gr. 

EXAMPLES. 

1.  Eeduce  5  yd.  2  ft.  7  in.  to  inches. 

2.  Reduce  27  gal.  2  qt.  1  pt.  3  gi.  to  gills. 

3.  Reduce  8  bu.  3  pk.  4  qt.  1  pt.  to  pints. 

4.  Reduce  29  cu.  yd.  8  cu.  ft.  999  cu.  in.  to  cubic  inches. 

5.  Reduce  145°  6'  33"  to  seconds. 

6.  Reduce  £24  18  s.  9  d.  2  far.  to  farthings. 

7.  Reduce  19  lb.  6  oz.  3  pwt.  20  gr.  to  grains. 

8.  Reduce  11  lb.  4  5  4  3  1  3  15  gr.  to  grains. 

9.  Reduce  4  T.  2  cwt.  1  qr.  11  lb.  to  ounces. 

10.  Reduce  3  0.  7  f  5  4  f  3  40  rn^  to  minims. 

11.  Reduce  8  cd.  2  cd.  ft.  13  cu.  ft.  to  cubic  feet. 

12.  Reduce  2  mi.  51  rd.  4  yd.  2  ft.  7  in.  to  inches. 

13.  Reduce  8  mi.  3  fur.  15  rd.  1  ft.  9  in.  to  inches. 


COMPOUND  NUMBERS.  83 

14.  Keduce  5  A.  101  sq.  rd.  25  sq.  yd.  112  sq.  in.  to  square 
inches. 

15.  Reduce  8  A.  2  R.  21  P.  17  sq.  yd.  6  sq.  ft.  89  sq.  in. 
to  square  inches. 

16.  Reduce  4  mi.  65  ch.  72  li.  5  in.  to  inches. 

.  17.   Reduce  3  yr.  7  wk.  6  da.  21  hr.  to  seconds. 

18.  Reduce  11  yr.  3  wk.  4  da.  18  hr.  to  minutes,  allowing 
for  three  leap  years. 

19.  How  many  units  are  there  in  8  gro.  8  doz.  ? 

20.  How  many  sheets  are  there  in  2  bundles  1  ream  15 
quires  10  sheets  ? 

21.  Find  the  number  of  ounces  in  a  long  ton. 

22.  Find  the  number  of  gills  of  molasses  in  a  barrel 
which  contains  86  gal. 

23.  What  is  the  value  of  a  silver  cup  weighing  10  oz. 
16  pwt.  at  12^  cents  a  pennyweight  ? 

24.  What  is  the  value  of  50  lb.  8  oz.  of  gold  at  $20.59J 
per  ounce  ? 

Reduction  Ascending. 

65.  When  a  compound  number  is  reduced  from  a  lower 
to  a  higher  denomination,  the  process  is  called  reduction 
ascending. 

I.   Reduce  766  gi.  to  higher  denominations. 

4)766  gi.  Since  there  are  4  gi.  in  1  pt.,  in 

2)191  pt.   2  si.  "^^^  S^-  t^^re  are  as  many  pints  as 

A\qn  nt    1  -nt  ^  ^'^  contained  times  in  766,  which 

—^      ,   o     1.  equals  191  pt  and  2  gi.  remaining. 

Z6  gal.  6  qt.  rj,^^^^  ^^g  2  pt.  in  1  qt. ;  in  191  pt. 

Ans.  23  gal.  3  qt.  1  pt.  2  gi.     there  are  as  many  quarts  as  2  is 

contained    times    in    191,    which 
equals  96  qt.  and  1  pt.  remaining.     There  are  4  qt.  in  1  gal. ;  in  96  qt. 


84  ARITHMETIC. 

there  are  as  many  gallons  as  4  is  contained  times  in  95,  which  equals 
23  gal.  and  3  qt.  remaining.     The  entire  result  is  23  gal.  3  qt.  1  pt.  2  gi. 

II.   Eeduce  104037  in.  to  higher  denominations. 

12)  104037  in.  The  method  is  the  same  as  that 

3)8669  ft.  9  in.  ^^®^  ^^  *^®   preceding  example. 

KixoSftQ  vd    2  ft  When  the  divisor  is  5|,  both  divi- 

o  2  dend  and  divisor  are  multiplied  by 

2  to  avoid  fractions.      The  divi- 


^ dend  and  divisor  thus  obtained  are 

320)525rd.f  yd.  =  1J yd.  half-yards;    hence  the  remainder 
1  mi.  205  rd.  is  3  half -yards,  which  equals  1^ 

1  mi.  205  rd.  1  yd.  2  ft.  9  in.   ^^ 


1  ft.  6  in. 


1  mi.  205  rd.  2  yd.  1  ft.  3  in. 


added  to  the  rest  of  the  answer. 

The  sum  of  9  in.  and  6  in.  is  15 
in.,  which  equals  1  ft.  3  in.  Write  the  3  in.,  and  carry  1  ft.  to  the 
column  of  feet.  The  sum  of  2  ft.,  1  ft.,  and  1  ft.  is  4  ft.,  which  equals 
1  yd.  1  ft.  Write  the  1  ft.  and  carry  1  yd.  to  the  column  of  yards.  The 
sum  of  1  yd.  and  1  yd.  is  2  yd. 

Note,  In  square  measure,  when  the  divisor  is  30|,  multiply  both 
dividend  and  divisor  by  4. 

III.    Reduce  4840371  min.  to  higher  denominations. 

60)4840371  min.  When   the   subject  of  time  is 

24)80672  hr.  51  min.  being  considered,  proper  allowance 

oaK^ZooF^  -,      o  1,  must   be    made    for    leap    j^ears. 
365)3361  da.  8  hr.  ^.  .     ^,  K  \ 
—                                          Smce  every  fourth  year  is  a  leap 

o  year,  in  9  years  there  are  at  least 

—  2  leap  years,  and  hence  2  da.  must 

•  ^  d^'  be  taken  from  the  76  da.  remain- 

Ans.  9  yr.  74  da.  8  hr.  51  min.  ^»g'  ^^^^h  leaves  74  da. 

In  order  to  ensure  absolute  ac- 
curacy with  regard  to  the  number  of  days,  the  exact  number  of  leap 
years  in  the  given  time  must  be  known. 

Note.  When  the  divisor  is  a  large  number,  it  is  more  convenient  to 
perform  the  long  division  at  one  side  of  the  work  and  then  tabulate 
the  results  as  if  it  had  all  been  done  by  short  division. 


COMPOUND  NUMBERS.  85 

EXAMPLES. 

1.  Eeduce  34718  far.  to  higher  denominations. 

2.  Eeduce  2763  gi.  to  higher  denominations. 

3.  Eeduce  935923  cu.  in.  to  higher  denominations. 

4.  Eeduce  67421"  to  higher  denominations. 

5.  Eeduce  49328  rri  to  higher  denominations. 

6.  Eeduce  677653  in.  to  higher  denominations. 

7.  Eeduce  10075  li.  to  higher  denominations. 

8.  Eeduce  147655  sq.  yd.  to  higher  denominations. 

9.  Eeduce  1286  pt.  to  bu.,  pk.,  etc. 

10.  Eeduce  54321  gr.  to  lb.,  oz.,  etc.  (Troy  Weight). 

11.  Eeduce  87634  gr.  to  lb.,  S,  etc. 

12.  How  many  cords   and  cord  feet  are  there  in  2224 
cubic  feet  ? 

13.  How  many  bales,  bundles,  etc.,  are  there  in  10379 
sheets  ? 

14.  Eeduce  8256120  sec.  to  higher  denominations. 

15.  In  372483  oz.  how  many  T.,  cwt.,  qr.,  etc.  ? 

16.  Change  106760  ft.  to  mi.,  rd.,  etc. 

17.  Eeduce  8868097  sq.  ft.  to  A.,  E.,  P.,  etc. 

18.  In  8476321  in.  how  many  mi.,  fur.,  rd.,  etc.  ? 

19.  In  1320765  sq.  in.  how  many  sq.  rd.,  sq.  yd.,  etc.  ? 

20.  In  80937864  sq.  in.  how  many  A.,  sq.  rd.,  etc.? 

21.  Eeduce  7963721  min.  to  yr.,  da.,  etc. 

22.  A  box  contains  12579  buttons ;  find  the  number  esti- 
mated in  great  gross,  gross,  etc. 


4rd. 

^yd. 

1ft. 

10  in. 

6 

0 

2 

7 

14 

2 

1 

9 

21 

3 

2 

6 

46 

3i 

2 

8 

1 

6 

86  ARITHMETIC. 


Addition  of  Compound  Numbers. 

66.   I.   Find  the  sum  of  4  rd.  2  yd.  1  ft.  10  in.,  6  rd.  2  ft. 
7  in.,  14  rd.  2  yd.  1  ft.  9  in.,  and  21  rd.  3  yd.  2  ft.  6  in. 

Since  only  units  of  the  same  kind 
can  be  added,  write  the  numbers  so 
that  units  of  the  same  kind  shall  be 
in  the  same  column.  Then  begin  at 
the  riglit  to  add.  The  sum  of  the 
inches  is  32  in.,  which  equals  2  ft. 
8  in.  Write  the  8  in.,  and  carry  2  ft. 
46  rd.     4  yd.     1  ft.      2  in.       to  the  column  of  feet.     The  sum  of 

the  feet,  including  2  ft.  previously 
obtained,  is  8  ft.,  which  equals  2  yd.  2  ft.  Write  the  2  ft.,  and  carry  2  yd. 
to  the  column  of  yards.  The  sum  of  the  yards,  including  2  yd.  previ- 
ously obtained,  is  9  yd.,  which  equals  1  rd.  3|  yd.  Write  the  3^  yd., 
and  carry  1  rd.  to  the  column  of  rods.  The  sum  of  the  rods,  including 
1  rd.  previously  obtained,  is  46  rd.  Cross  out  the  |  yd.,  and  write  its 
equivalent  1  ft.  6  in.  Then  add  again,  and  the  entire  result  is  46  rd. 
4  yd.  1  ft.  2  in. 

EXAMPLES. 

1.  Find  the  sum  of  18  gal.  3  qt.  1  pt.  1  gi.,  5  gal.  1  pt. 
3  gi.,  16  gal.  2  qt.  2  gi.,  and  4  gal.  1  qt.  1  pt. 

2.  Find  the  sum  of  101  bu.  3  pk.  5  qt.,  27  bu.  2  pk.  6  qt. 

1  pt.,  14  bu.  1  qt.  1  pt.,  and  33  bu.  3  pk.  7  qt. 

3.  Find  the  sum  of  27°  30'  54",  32°  24'  58",  62°  47'  25", 
and  75°  29'  47". 

4.  Find  the  sum  of  16  cd.  6  cd.  ft.  14  cu.  ft.,  22  cd.  2  cd. 
ft.  4  cu.  ft.,  and  19  cd.  1  cd.  ft.  2  cu.  ft. 

5.  Find  the  sum  of  74  cu.  yd.  20  cu.  ft.  918  cu.  in.,  29  cu. 
yd.  15  cu.  ft.  1000  cu.  in.,  and  14  cu.  yd.  2  cu.  ft.  323  cu.  in. 

6.  Find  the  sum  of  4  wk.  2  da.  17  hr.  48  min.  37  sec,  6  da. 

2  hr.  29  min.  13  sec,  and  12  wk.  1  da.  11  hr.  16  min.  4  sec 


COMPOUND   NUMBERS.  87 

7.  Find  the  sum  of  £5  15  s.  6  d.  2  far.,  £7  8  s.  1  far.,  19  s. 
7d.  3  far.,  and  £12  10  s. 

8.  Find  the  sum  of  3  lb.  9  oz.  15  pwt.  12  gr.,  6  lb.  16 
pwt.  8  gr.,  and  3  lb.  11  oz.  7  pwt.  4  gr. 

9.  Eind  the  sum  of  5  lb.  9  5   6  3  2  3  15  gr.,  7  lb.  11  5 
73238  gr.,  and  11  lb.  7  5  G  3  1  3  3  gr. 

10.  Find  the  sum  of  18  T.  7  cwt.  1  qr..  18  lb.  12  oz.,  18 
cwt.  3  qr.  21  lb.  6  oz.,  and  9  T.  14  cwt.  15  lb.  15  oz. 

11.  Find  the  sum  of  5  T.  18  cwt.  52  lb.  8  oz.  6  dr.,  15  T. 
7  cwt.  44  lb.  10  oz.  12  dr.,  and  15  cwt.  78  lb.  12  oz.  14  dr. 

12.  Find  the  sum  of  3  mi.  7  fur.  19  rd.  4  yd,  1  ft.,  7  mi. 
1  fur.  32  rd.  1  yd.  2  ft.,  and  9  mi.  6  fur.  25  rd.  3  yd. 

13.  Find  the  sum  of  15  mi.  110  rd.  4  yd.  1  ft.  6  in.,  22  mi. 
15  rd.  5  yd.  2  ft.  10  in.,  17  mi.  214  rd.  2  yd.  7  in.,  and  63  rd. 

I  yd.  1  ft.  11  in. 

14.  Find  the  sum  of  28  A.  120  sq.  rd.  15  sq.  yd.  7  sq.  ft. 
120  sq.  in.,  6  A.  91  sq.  rd.  21  sq.  yd.  4  sq.  ft.  32  sq.  in.,  and 
65  A.  11  sq.  rd.  12  sq.  yd.  6  sq.  ft.  14  sq.  in. 

15.  Find  the  sum  of  3  A.  2  R.  17  P.  22  sq.  yd.  8  sq.  ft. 
28  sq.  in.,  4  A.  1  R.  35  P.  17  sq.  yd.  4  sq.  ft.  92  sq.  in.,  and 

II  A.  26  P.  26  sq.  yd.  7  sq,  ft.  116  sq.  in. 

Subtraction  of  Compound  Numbers. 

67.  I.  Subtract  7  T.  13  cwt.  78  lb.  5  oz.  from  18  T.  6  cwt. 
31  lb.  12  oz. 

18  T.       6  cwt.  31  lb.    12  oz.  Write  the  subtrahend  under  the 

7  13  78  5  minuend  so  that  units  of  the  same 

TTZZ      TZ        ]     ZZT-,         Z  kind  shall  be  in  the  same  column. 

10  T.    12  cwt.  53  lb.     7  oz.     rp,      ,     .      ^  ^,      .  w  ^       ,  ^     ^ 

Then  begm  at  the  right  to  subtract. 

Subtracting  5  oz.  from  12  oz.,  we  have  7  oz.  remaining.     We  cannot 

subtract  78  lb.  from  31  lb.,  so  we  take  1  cwt.  from  6  cwt.  and  add  it. 


88  AEITHMEO'lC. 

reduced  to  pounds,  to  31  lb.,  making  131  lb. ;  subtracting  78  lb.  from 
131  lb.,  we  have  53  lb.  We  cannot  subtract  13  cwt.  from  5  cwt. ;  so  we 
take  1  T.  from  18  T.  and  add  it,  reduced  to  hundred-weight,  to  5  cwt., 
making  25  cwt. ;  subtracting  13  cwt.  from  25  cwt.,  we  have  12  cwt. 
Subtracting  7  T.  from  17  T.,  we  have  10  T.  The  entire  result  is  10  T. 
12  cwt.  53  lb.  7  oz. 

EXAMPLES. 

1.  Subtract  8  cwt.  72  lb.  9  oz.  from  13  cwt.  90  lb.  14  oz. 

2.  Subtract  5  cd.  6  cd.  ft.  8  cu.  ft.  from  76  cd.  3  cd.  ft.  12 
cu.  ft. 

3.  Subtract  35  gal.  3  qt.  2  gi.  from  57  gal.  2  qt.  1  pt.  1  gi. 

4.  Subtract  2  bu.  3  pk.  2  qt.  1  pt.  from  7  bu.  3  pk. 

5.  Subtract  £13  12s.  5d.  3far.  from  £21  9s.  7d.  2far. 

6.  Subtract  54°  42'  30"  from  90°. 

7.  Subtract  12  lb.  5  oz.  16  pwt.  15  gr.  from  18  lb.  6  oz. 
14  pwt.  3  gr. 

8.  Subtract  9  lb.  10  5  5  3  1  3  16  gr.  from  12  tb.  8  5  4  3 
2  3  7  gr. 

9.  Subtract  10  lb.  8  oz.  from  5  T. 

10.  Subtract  2  yr.  202  da.  21  hr.  54  min.  25  sec.  from  5  yr. 
71  da.  15  hr.  45  min.  45  sec. 

11.  Subtract  3  da.  16  hr.  18  min.  45  sec.  from  1  wk. 

12.  Subtract  4  mi.  220  rd.  4  yd.  2  ft.  7  in.  from  5  mi.  115 
rd.  2  yd. 

13.  Subtract  6  fur.  32  rd.  5  yd.  1  ft.  6  in.  from  3  mi.  4  fur. 
32  rd.  3  yd.  10  in. 

14.  Subtract  7  sq.  yd.  139  sq.  in.  from  1  sq.  rd.  5  sq.  ft. 

16.   Subtract  3  A.  3  E.  12  P.  18  sq.  yd.  8  sq.  ft.  87  sq.  iiL 
from  4  A.  2  E.  16  P.  11  sq.  yd.  7  sq.  ft.  54  sq.  in. 

16.   What  is  the  difference  between  2S  mi.  and  27  mi.  7 
fur.  39  rd.  5  ft.  11.9  in.? 


COMPOUND   NUMBERS.  89 


Difference  between  Dates. 

68.  I.  Find  the  difference  of  time  between  Aug.  25th, 
1872  and  Mar.  12th,  1887. 

1887  —  3  —  12  In  subtracting  dates  it  is  customary  to 

1872  —  8  —  25  reckon  30  da.  to  a  month.      Since  March 

— ■        —      and  August  are  respectively  the  third  and 

14  yr.  6  mo.  17  da.  ^j^j^^,^  months,  we  write  3  and  8  instead  of 
the  names  of  the  months.  The  subtraction  is  then  performed  like  that 
of  ordinary  compound  numbers. 

II.  Find  the  exact  number  of  days  from  Dec.  23rd,  1887 
to  Apr.  13th,  1888. 

When  the  exact  number  of  days  is  wanted,  we  must  take 
the  actual  number  of  days  in  eacli  month.  We  must  not 
count  in  both  dates;  it  is  customary  to  omit  the  former  and 
count  the  latter.  From  Doc.  23rd  to  the  end  of  the  month 
there  are  8  da.  January  has  31  da.  Since  1888  is  a  leap 
year,  February  has  29  da.  March  has  31  da.  There  are  13 
112  da.  da.  to  be  counted  in  April.  By  addition  we  find  the  result 
to  be  112  da. 


8 
31 
29 
31 
13 


EXAMPLES. 

1.  Find  the  difference  of  time  laetween  Feb.  18th,  1856 
and  Oct.  25th,  1874. 

2.  Find  the  difference  of  time  between  Nov.  19th,  1874 
and  Apr.  Srd,  1888. 

3.  Find  the  difference  of  time  between  Aug.  31st,  1876 
and  May  10th,  1886. 

4.  Washington  was  born  Feb.  22nd,  1732,  and  died  Dec. 
14th,  1799 ;  what  was  his  age  ? 

5.  A  ship  started  on  a  whaling  voyage  Apr.  24th,  1875, 
and  returned  Sept.  8th,  1880  ;  how  long  was  it  gone  ? 


90  ARITHMETIC. 

6.  A  mortgage  was  dated  Oct.  21st,  1879,  and  was  paid 
July  1st,  1887  ;  how  lon^  did  it  run  ? 

7.  On  the  1st  of  January,  1888,  how  much  time  had 
passed  since  the  Declaration  of  Independence,  July  4th, 
1776? 

8.  Find  the  exact  number  of  days  from 'Apr.  5th,  1883 
to  Dec.  9th,  1883. 

9.  Find  the  exact  number  of  days  from  Aug.  19th,  1883 
to  June  25th,  1884. 

10.  Find  the  exact  number  of  days  from  May  14th,  1884 
to  Nov.  5th,  1884. 

11.  Find  the  exact  number  of  days  from  Dec.  27th,  1884 
to  July  5th,  1885. 

12.  A  note  was  dated  July  30th,  1886,  and  was  paid  May 
9th,  1887  ;  how  many  days  did  it  run  ? 

13.  A  man  started  on  a  business  trip  Kov.  14th,  1887,  and 
returned  Mar.  9th,  1888 ;  how  many  days  was  he  gone  ? 

14.  If  the  spring  term  of  a  school  ends  June  22nd,  and 
the  fall  term  begins  Sept.  19th,  how  many  days  are  there  in 
the  summer  vacation  ? 

Multiplication  of  Compound  Numbers. 

69.   I.   Multiply  £4  13  s.  10 d.  3  far.  by  7. 

£4     13  s.  10  d.  3  far.  Write  the  multiplier  under  the  lowest 

7  denomination  of  the  multiplicand,  and  be- 

£32  17s.  3d.  Ifar.  S^^  **  *^^^  right  to  multiply.  7  times  3 far. 
are  21  far.,  which  equals  5  d.  1  far.  Write 
the  1  far.,  and  reserve  5  d,  to  be  added  to  the  product  of  the  pence. 
7  times  10  d.  are  70  d. ;  the  sum  of  70  d.  and  5  d.  is  75  d.,  which  equals 
6  s.  3  d.  Write  the  3  d.,  and  reserve  6  s.  to  be  added  to  the  product  of 
the  shillings.    7  times  13  s.  are  91  s. ;  the  sum  of  91  s.  and  6  s.  is  97  s.. 


COMPOUND  NUMBERS.  91 

which  equals  £4  17  s.  Write  the  17  s.,  and  reserve  £4  to  be  added  to 
the  product  of  the  pounds.  7  times  £4  are  £28 ;  the  sum  of  £28  and 
£4  is  £32.    The  entire  result  is  £32  17  s.  3  d.  1  far. 

EXAMPLES. 

1.  Multiply  12  T.  14cwt.  Iqr.  181b.  10  oz.  by  4. 

2.  Multiply  4  yr.  78  da.  18  hr.  15  min.  30  sec.  by  6. 

3.  Multiply  8  A.  2  R.  22  P.  6  sq.  yd.  5  sq.  ft.  42  sq.  in.  by  8. 

4.  Multiply  18  bu.  3pk.  5qt.  1  pt.  by  9. 

5.  Multiply  3  A.  104  sq.  rd.  25  sq.  yd.  8  sq.  ft.  by  12. 

6.  Multiply  37  gal.  3  qt.  1  pt.  2  gi.  by  14. 
-    7.  Multiply  24°  36' 50"  by  15. 

8.  Multiply  15  cu.  yd.  12  cu.  ft.  227  cu.  in  by  18. 

9.  Multiply  21  tb.  9  5  2  3  1  3  16  gr.  by  20. 

10.  Multiply  15  mi.  128  rd.  1  ft.  by  23. 

11.  Multiply  £9  17  s.  6d.  1  far.  by  28. 

12.  Multiply  5  T.  8  cwt.  64  lb.  8  oz.  6  dr.  by  37. 

13.  Multiply  8  mi.  5  fur.  16  rd.  3  yd.  2  ft.  8  in.  by  48. 

14.  Multiply  4  lb.  8  oz.  16  pwt.  20  gr.  by  72. 

Division  of  Compound  Numbers. 

70.  I.   Divide  41  bu.  1  pk.  7  qt.  1  pt.  by  9. 
9)41  bu.  1  pk.  7  qt.  1  pt.  Write  the  divisor  at  the  left  of  the 

4  bu.  2  pk.  3  qt.  1  pt.  dividend,  and  begin  at  the  left  to  divide. 
41  bu.  divided  by  9  equals  4  bu.,  with  a 
remainder  of  ^  bu.  Write  the  4  bu.,  and  reduce  5  bu.  to  pecks;  the  re- 
sult, after  adding  1  pk.,  is  21  pk.  21  pk.  divided  by  9  equals  2  pk.,  with 
a  remainder  of  3  pk.  Write  the  2  pk.,  and  reduce  3  pk.  to  quarts;  the 
result,  after  adding  7  qt.,  is  31  qt.  31  qt.  divided  by  9  equals  3  qt., 
with  a  remainder  of  4  qt.  Write  the  3  qt.,  and  reduce  4  qt.  to  pints; 
the  result,  after  adding  1  pt.,  is  9  pt.  9  pt.  divided  by  9  equals  1  pt 
The  entire  result  is  4  bu.  2  pk.  3  qt.  1  pt. 


92  ARITHMETIC. 

II.  Divide  161  cd.  4  cd.  ft.  13  cu.  ft.  by  21. 

21)161  cd,  4  cd.  ft.  13  cu.  ft.  (7  cd. 

147 

14 

_8 

112 

4 

?l)116cd.ft.  (5cd.ft.  The  arrangement  of    work 

_    '^  here  given  can  be  used  when 

11  the  divisor  is  a  large  number. 
J6 

176 

J3  .  . 

21)189(9  cu.  ft. 
189 

Ans.  7  cd.  5  cd.  ft.  9  cu.  ft. 

III.  Divide  245°  34'  12"  by  9°  26'  42". 


9°  26'  42" 

245°  34' 

12" 

60 

60 

540 

14700 

34002)884052(26 

26 

34 

68004 

566' 

14734' 

204012 

60 

60 

204012 

33960 

884040 

42 

12 

34002"         884052" 

When  both  dividend  and  divisor  are  compound  numbers,  reduce 
both  to  the  lowest  denomination  mentioned  in  either,  and  divide  as  in 
simple  numbers.    Notice  tliat  the  answer  is  26,  not  26". 

EXAMPLES. 

1.  Divide  3  wk.  6  da.  14  hr.  17  min.  57  sec.  by  3. 

2.  Divide  38  A.  114  sq.  rd.  11  sq.  yd.  4  sq.  ft.  72  sq. 
in.  by  5. 


COMPOU]ST>  NUMBERS.  93 

3.  Divide  £32  16  s.  3  d.  by  7. 

4.  Divide  5  cwt.  12  lb.  4  oz.  by  7. 

5.  Divide  85  gal.  2  qt.  1  pt.  3  gi.  by  11. 

6.  Divide  32  lb.  10  oz.  7  pwt.  18  gr.  by  13. 

7.  Divide  4°  3'  17".06  by  15. 

8.  Divide  347  bu.  1  pk.  7  qt.  1  pt.  by  19. 

9.  Divide  6  mi.  7  fur.  30  rd.  2  ft.  by  48. 

10.  Divide  152  lb.  5  5  1  3  2  3  14  gr.  by  61. 

11.  Divide  3  rd.  3  yd.  2  ft.  10  in.  by  2  ft.  8  in. 

12.  Divide  35  gal.  3  qt.  1  pt.  1  gi.  by  4  qt.  1  pt.  3  gi. 

13.  Divide  15  cwt.  2  qr.  19  lb.  12  oz..  by  1  qr.  12  lb.  6  oz. 

14.  Divide  92  cd.  1  cd.  ft.  by  5  cd.  5  cd.  ft.  6  cu.  ft. 

15.  Divide  £131  Is.  lOd.  2  far.  by  £8  14s.  9d.  2  far. 

16.  How  many  house  lots,  each  containing  1  A.  89  sq.  rd., 
are  there  in  a  piece  of  land  containing  56  A.  4  sq.  rd.  ? 

17.  How  many  coins,  each  weighing  15  pwt.  18  gr.,  can 
be  made  out  of  16  lb.  10  oz.  7  pwt.  18  gr.  of  metal  ? 

18.  How  many  bags,  each  containing  2  bu.  1  pk.  3  qt., 
will  be  required  to  hold  111  bu.  2  pk.  4  qt.  of  grain  ? 


To  Multiply  or   Divide  a  Compound  Number   by  a 
Fraction. 


71.  I.   Multiply  4  gal.  1  pt.  3  gi.  by  f . 

4  gal.  0  qt.  1  pt.  3  gi. 


3  Multiplying  by  |  is  the  same  as 


5)12  gal.  2qt.  1  pt.  1  gi.  multiplying  by  3  and  dividing  the 

2gal.  2qt.  Igi.  result  by  6. 


94  ARITHMETIC. 

To  multiply  a  compound  number  by  a  fraction,  multiply 
by  the  numerator  and  divide  by  the  denominator. 

If  the  multiplier  is  a  mixed  number,  multiply  by  the 
integral  and  fractional  parts  separately,  and  add  the  results. 

II.   Divide  2  lb.  8  oz.  2  pwt.  6  gr.  by  ^. 

2  lb.  8  oz.  2  pwt.    6  gr. 

9  Dividing  by  |  is  the  same  as  mul- 


7)24  lb.  1  oz.  0  pwt.    6  gr.      tiplying  by  f . 
3  lb.  5  oz.  5  pwt.  18  gr. 

To  divide  a  compound  number  by  a  fraction,  midtiply  by 
the  denominator  and  divide  by  the  numerator. 

If  the  divisor  is  a  mixed  number,  reduce  the  mixed  num- 
ber to  an  improper  fraction,  and  proceed  as  before. 

EXAMPLES. 

1.  Multiply  5  T.  6  cwt.  48  lb.  5  oz.  by  f  . 

2.  Multiply  13  cd.  2  cd.  ft.  9  cu.  ft.  by  ^. 

3.  Multiply  7  bu.  2  pk.  3  qt.  1  pt.  by  j\. 

4.  Multiply  14  lb.  8  5  5  3  1  3  12  gr.  by  ff . 

5.  Multiply  4  mi.  112  rd.  1  yd.  1  ft.  8  in.  by  5|. 

6.  Multiply  23°  6'  41"  by  13^. 

7.  Multiply  2  wk.  4  da.  19  hr.  7  min.  24  sec.  by  8JJ. 

8.  Divide  £4  14s.  3d.  3  far. by  j%. 

9.  Divide  7  sq.  rd.  4  sq.  yd.  6  sq.  ft.  41  sq.  in.  by  ^. 

10.  Divide  23  gal.  3  qt.  1  pt.  3  gi.  by  if. 

11.  Divide  2  mi.  3  fur.  16  rd.  2  yd.  2  ft.  6  in.  by  3f 

12.  Divide  13  T.  13  cwt.  3  qr.  9  lb.  6  oz.  by  8|. 

13.  Divide  14°  58'  8"  by  ll^^. 


COMPOUND  NUMBEh-S.  95 


To  KEDX;t;i!,'  A  FRACTION  OF  ONE  DENOMINATION  TO    LoWER 

Denominations. 

72.  I.   Keduce  -^  of  a  yard  to  the  fraction  of  an  inch. 

-  «  3  times  the  number  of  yards  equals  the  num- 

—  X3xX?=Ti>i'  ber  of  feet,  and  12  times  the  number  of  feet 

4  equals  the  number  of   inches.     Hence  :^  yd. 

equals  jV  X  3  X  12  in.,  which  equals  f  in. 

II.   Kednce  ^  cwt.  to  lower  denominations. 

7      25     ,-. 

±XX^0  =  ^  =  29^  lb.  Since  there  are  100  lb.  in  1  cwt,  in 

^^  ^:i  cwt.  there  are  ^^^  of  100  lb.,  which 

equals  29^  lb.    In  1  lb.  there  are  16  oz. ; 

i  X  10  =  -  =  22  oz  ^"  s  ^^*  *^'®'*^  ^^  i  ®^  1^  ®2''  which  equals 

^  3  2§  oz.     In  1  oz.  there  are  16  dr. ;  in  jj  oz. 

^  there  are  i?  of  16  dr.,  which  equals  lOJ 

?Xl6  =  — =  10f  dr.  dr.     The  entire  result  is  29  lb.  2  oz. 

3  3  lOfdr. 

Ans.  291b.2oz.l0|dr. 

When  fractions  of  different  denominations  are  to  be  added 
or  subtracted,  reduce  the  fractions  to  lower  denominations, 
and  then  perform  the  operations  indicated. 


EXAMPLES. 

1.  Eeduce  ^ro-  ^^  ^  cord  to  the  fraction  of  a  cubic  foot. 

2.  Eeduce  -g^^  of  a  shilling  to  the  fraction  of  a  farthing. 

3.  Keduce  -^  of  a  gallon  to  the  fraction  of  a  gill. 

4.  Eeduce  ^  of  a  bushel  to  the  fraction  of  a  pint 

5.  Reduce  -ff  of  a  week  to  minutes. 

6.  Reduce  ^^  of  a  furlong  to  inches. 

7.  Reduce  -^^  of  a  pound  Troy  to  grains. 

8.  Reduce  y|^  of  a  rood  to  square  feet. 


96  AEITHMETIC. 

9.  Eeduce  ^  bu.  to  lower  denominations. 

10.  Keduce  f  tb.  to  lower  denominations. 

11.  Reduce  f  rd.  to  lower  denominations. 

12.  Reduce  -^  mi.  to  lower  denominations. 

13.  Reduce  i  yr.  to  lower  denominations. 

14.  Reduce  ^^  bu.  to  lower  denominations. 

15.  Reduce  -f^  A.  to  lower  denominations. 

16.  Reduce  ^  A.  to  lower  denominations. 

17.  Reduce  ff  gal.  to  lower  denominations. 

34 

18.  Find  the  value  of  -^  of  J  of  an  acre  at  $1.36  per 

square  foot.  ^'^ 

19.  Add  ^  of  a  furlong,  ^  of  a  rod,  and  ^  of  a  yard. 

20.  From  f  of  a  gallon  subtract  1-|  of  a  pint. 

21.  Add  |-  of  a  pound,  |  of  a  shilling,  and  -f-  of  a  penny. 

22.  Add  -^  lb.,  m  oz.,  and  9|-  pwt. 

23.  Express  in  rods,  yards,  etc.,  -f-^  mi.  +|  of  40  rd.  +  |yd. 

24.  From  |  of  4|  bu:  subtract  f  of  2^  pk. 

25.  Add  together  f  of  £13,  |  of  i-  of  f  of  £2  12  s.,  and 
fof9d.  ^^ 

To  Reduce  Lower  Denominations  to  a  Fraction 
OF  A  Higher  Denomination. 

73.   I.    Reduce  -|  of  a  grain  to  the  fraction  of  an  ounce 
Troy. 

^1  y  J_  _  _1_  ^V  ^f  *^®  number  of  grains  equals  the 

3      24      ^0      720      '  number  of  pennyweights,  and   ^V  of  the 

1^  number  of  pennyweights  equals  the  num- 

ber of  ounces.    Hence  f  gr.  equals  f  X  ^V ^ ^V  o*=>  which  equals  j^-^ oz. 


20  _ 

=  lhr. 

60" 

3 

2 

M- 

=  ^X 

1 

_2 

da. 

24 

3 

3 

""  9 

2|  = 

=  '^x 

1_ 

20 

wk. 

COMPOUND  KUMBEKS.  97 

IL   KeduL'e  ij?  da.  5  hr,  20  min.  to  the  fraction  of  a  week. 

Since  there  are  60  min.  in  1  iir.,  20  min. 
equal  |g  of  an  hour,  whioii  equals  \  hr. 
In  1  da.  there  are  24  hr. ;  5^  hr.  equals 
—  of  a  day,  which  equals  f  da.  In  1  wk. 
there  are  7  da.;  2f  da.  e^ual  -2  of  a  week 
9"  "^  7  ~  63  ""^  which  equals  l^  wk. 

EXAMPLES. 

1.  Keduce  ^  of  a  gill  to  the  fraction  of  a  gallon. 

2.  Keduce  ^  of  an  inch  to  the  fraction  of  a  rod. 

3.  Reduce  ^  of  a  pound  to  the  fraction  of  a  ton. 

4.  Reduce  2|  qt.  to  the  fraction  of  a  bushel. 

5.  Reduce  9^  sq.  ft.  to  the  fraction  of  a  rood. 

6.  What  part  of  a  mile  is  one  inch  ? 

7.  Reduce  12  s.  6d.  to  the  fraction  of  a  pound. 

8.  Reduce  9  hr.  20  min.  to  the  fraction  of  a  week. 

9.  Reduce  7  qt.  1 J  pt.  to  the  fraction  of  a  bushel. 

10.  Reduce  9  rd.  1  ft.  6  in.  to  the  fraction  of  a  furlo/ig. 

11.  Reduce  6  cwt.  2  qr.  24  lb.  to  the  fraction  of  a  ton. 

12.  Reduce  15  5  3  1  3  16  gr.  to  the  fraction  of  a  pound. 

13.  Reduce  6  rd.  5  ft.  9  in.  to  the  fraction  of  a  mile. 

14.  Reduce  40  sq.  rd.  27  sq.  yd.  4  sq.  ft.  72  sq.  in.  to  the 
fraction  of  an  acre. 

15.  Reduce  3  bundles  9  quires  4  sheets  to  the  fraction  of 
a  bale. 

16.  What  part  of  a  hogshead  is  3  gal.  1  qt.  2^  gi.  ? 


98  ARITHMETIC. 

To  Find  what  Fractional  Part  one  Compound 
Number  is  of  Another. 

74.   I.  What  part  of  32  bu.  2  pk.  4  qt.  is  8  bu.  3  pk.  2  qt.  ? 

8     2^  8     4^ 

21     5  31      13                Reduce  both  compound  numbers  to  the 

4  ~  §      ■  4  ~  16  same  denomination,  and  then  by  the  method 

.«  shown  in  §  41,  find  what  fractional  part  one 

814     X^X  ^        47  number  is  of  the  other. 

2        87 

EXAMPLES. 

1.  What  part  of  6  A.  1  E.  is  3  E.  5  P.  ? 

2.  What  part  of  3  bu.  2  pk.  is  5  pk.  6  qt.  ? 

3.  What  part  of  51°  25'  20"  is  3°  51'  24"  ? 

4.  What  part  of  37  sq.  yd.  2  sq.  ft.  116  sq.  in.  is  10  sq.  yd. 

5  sq.  ft.  136  sq.  in.  ? 

5.  What  part  of  2  A.  71  sq.  rd.  is  1  A.  7  sq.  ch.  ? 

6.  What  part  of  4  da.  is  1  da.  9  hr.  13  min.  50f|  sec.  ? 

7.  What  part  of  11  mi.  156  rd.  5  yd.  is  1  mi.  69  rd.  1  ft. 

6  in.? 

8.  What  part  of  22  gal.  3  qt.  1  gi.  is  4  gal.  2  qt.  1  pt.  2  gi.? 

9.  What  part  of  4  lb.  9  5  1  3  12  gr.  is  10  5  636  gr.  ? 

10.  What  part  of  £18  15s.  4d.  2far.  is  £4  3s.  lOd.  2far.  ? 

11.  What  part  of  5  fur.  3  rd.  3  yd.  1  ft.  6  in.  is  4  fur.  17 
rd.  4  yd.  10  in.  ? 

12.  What  part  of  8  long  tons  is  |  of  720  lb.  ? 

/-» 2 

13.  What  part  of  -^  yards  is  |-  of  an  inch  ? 

TT 

14.  What  part  12  yd.  1  ft.  6  in.  is  -^^  of  a  mile  ? 


COMPOUND  NUMBERS.  99 

To  Eeduce  a  Decimal  of  one  Denomination  to 
Lower  Denominations. 

75.   I.   Reduce  0.4375  gal.  to  lower  denominations. 

0.4375  gal. 

4  Since  there  are  4  qt.  in  1  gal.,  in  0.4.375  gal. 

1.750P  qt.  there  are  0.4375  of  4  qt.,  which  equals  1.76  qt. 

^  In  1  qt.  there  are  2  pt. ;  in  0.75  qt.  there  are  0.75 

■'^•^^P*-  of  2pt.,  which  equals  1.5  pt.     In  1  pt.  there  are 

— — -  ,  4  gi. ;    in  0.5  pt.  there  are  0.5  of  4  gi.,  which 

■'^  ^^'  equals  2  gi.     The  entire  result  is  1  qt.  1  pt.  2  gi. 
Ans.  1  qt.  1  pt.  2  gi. 

When  decimals  of  different  denominations  are  to  be  added 
or  subtracted,  reduce  them  to  the  same  denomination,  and 
then  perform  the  operations  indicated. 

EXAMPLES. 

1.  Eeduce  0.7375  lb.  Troy  to  lower  denominations. 

2.  Eeduce  0.5625  da.  to  lower  denominations. 

3.  Eeduce  0.3125  bu.  to  lower  denominations. 

4.  Eeduce  0.4125  lb.  to  lower  denominations. 

5.  Eeduce  0.0625  A.  to  lower  denominations. 

6.  Eeduce  0.795  wk.  to  lower  denominations. 

7.  Eeduce  0.445  A.  to  lower. denominations. 

8.  Eeduce  0.845  mi.  to  lower  denominations. 

9.  Eeduce  0.428  T.  to  lower  denominations. 

10.  Eeduce  0.333  A.  to  lower  denominations. 

11.  Eeduce  0.984375  bu.  to  lower  denominations. 

12.  Eeduce  0.758762  A.  to  lower  denominations. 

13.  Subtract  10.869  oz.  from  1.203  lb.  Troy. 


100  ARITHMETIC. 

14.  Add  together  1.001  cwt.  and  0.039  qr.,  and  gire  fh^ 
answer  in  ounces  and  the  decimal  of  an  ounce. 

15.  Subtract  0.335  gal.  from  12.51  qt.,  and  give  the  answer 
in  pints  and  the  decimal  of  a  pint. 

16.  Subtract  7.3125  fur.  from  1.03125  mi.,  and  give  the 
answer  in  yards  and  the  decimal  of  a  yard. 

'  17.   What  is  the  cost  of  0.33  bbl.  of  wine  at  ^1.15  per  pint  ? 

18.    A  man  bought  a  piece  of  ground  containing  0.316  A. 
at  53  cents  a  square  foot ;  what  did  he  pay  for  the  piece  ? 

To  Eeduce  Lower  Denominations  to  a  Decimal  of  a 
Higher  Denomination. 

76.   I.   Reduce  7  5   7  3  1  3  16  gr.  to  the  decimal  of  a 
pound. 

Since  there   are  20  gr.  in  1  3,  the  number  of 

scruples  equals  2V  of  t^e  number  of  grains ;  ^^  of 

16  is  0.8,^ which,  added  to  1  3,  equals  1.8  3-    Since 

there  are  3  Q  in  1  3 ,  the  number  of  drams  equals 

\  of  the  number  of  scruples ;   I  of  1.8  is  0.6,  which, 

0  CC2^  ft       added  to  7  3  >  equals  7.6  3  •    Since  there  are  8  3  in 

1  5  J  the  number  of  ounces  equals  I  of  the  number 

of  drams;  1  of  7.6  is  0.95,  which,  added  to  7  5,  equals  7.95  §.      Since 

there  are  12  ^  in  1  lb.,  the  number  of  pounds  equals  ^^  of  the  number 

of  ounces ;  j\  of  7.95  is  0.6625. 

EXAMPLES. 

1.  Reduce  4  s.  9  d.  to  the  decimal  of  a  pound. 

2.  Reduce  5  yd.  2  ft.  6  in.  to  the  decimal  of  a  rod. 

3.  Reduce  3  R.  13  P.  8  sq.  ft.  to  the  decimal  of  an  acre. 

4.  Reduce  12  s.  9  d.  2  far.  to  the  decimal  of  a  pound. 

5.  Reduce  15  lb.  5  oz.  4  dr.  to  the  decimal  of  a  ton. 


20 

16.0  gr. 

3 

IS  3 

8 

7.60  3 

12 

7.9500  5 

COMPOUND  NUMBERS.  101 

6.  Eeduce  5  fur.  33  rd.  9  ft.  10.8  in.  to  the  decimal  of  a 
mile.  ',    '  ;- :  :    ,: 

7.  Keduce  2  cwt.  3  qr.  3  lb.  8  oz.  to  tl^e  decimal  of  i^.tjcn 

8.  Reduce  1  hr.  25  min.  30  sec.  to  the  decimal'  of  a  day. 

9.  Reduce  1  oz.  8  pwt.  19.2  gr.  to  the  decimal  of  a  pound. 

10.  Reduce  6  fur.  30  rd.  6  ft.  7.2  in.  to  the  decimal  of  a 
mile. 

11.  Reduce  38  sq.  rd.  21  sq.  yd.  5  sq.  ft.  108  sq.  in.  to  the 
decimal  of  an  acre. 

12.  Reduce  30  rd.  4  yd.  2  ft.  10  in.  to  the  decimal  of  a  mile. 

13.  Reduce  71  sq.  rd.  54  sq.  ft.  64.8  sq.  in.  to  the  decimal 
of  an  acre. 

14.  What  decimal  part  of  a  degree  is  52'  43".5  ? 

15.  Reduce  1  fur.  25  rd.  12  ft.  11  in.  to  the  decimal  of  a 
mile. 

16.  Express  to  the  nearest  millionth  34  A.  7  sq.  rd.  3  sq. 
yd.  7  sq.  ft.  104  sq.  in.  as  a  decimal  of  a  square  mile. 

17.  Reduce  2  yr.  5  mo.  12  da.  to  years  and  the  decimal  of 
a  year. 

18.  Reduce  18216  ft.  to  miles  and  the  decimal  of  a  mile. 

19.  Reduce  ^  of  a  farthing  to  the  decimal  of  a  pound. 

20.  At  6  cents  a  pound,  what  decimal  part  of  a  ton  of 
nails  can  be  bought  for  f  4.20  ? 

21.  Find  the  value  of  21  A.  3  R.  12  P.  of  land  at  $45 
per  acre. 

22.  Find  the  value  of  7  A.  35  sq.  rd,  127  sq.  ft.  of  land 
at  f  108.15  per  acre. 

23.  What  is   the  value,  at  $4500  an  acre,  of  a  piece  oi 
land  containing  30  sq.  rd.  19  sq.  ft.  89  sq.  in.  ? 


102  AKITHMETIO. 

To  FiND^  WHAT  Decimal  one  Compound   Number  is  op 
:«  ;    ./  Another. 

.:77:."  £   What,  decimal  of  8  bu.  3  pk.  is  2  bu.  1  pk.  5  qt.  ? 

8  bu.  3  pk.  2  bu.  1  pk.  5  qt. 

_4  4  280)77.000(0.275 

32  8  5^ 

_3  1  2100 

35  pk.  9pk.-  I960 

8  _8  •       1400 

280  qt.  72  1400 

_5 

77  qt. 
8  bu.  3  pk.  =  280  qt.     2  bu.  1  pk.  5  qt.  =  77  qt.     77  qt.  is  ^^^  of  280 
qt.,  and  this  can  be  expressed  as  a  decimal  by  dividing  77  by  280. 

Note.  It  is  merely  necessary  to  reduce  both  compound  numbers  to 
the  same  denomination,  and  then  divide.  Choose  the  denomination  in 
which  the  two  numbers  can  be  most  simply  expressed. 

EXAMPLES. 

1.  What  decimal  of  132  bu.  is  8  bu.  1  pk.  ? 

2.  What  decimal  of  2  gal.  is  1  qt:  1  pt.  2  gi.  ? 

3.  What  decimal  of  19  s.  6  d.  is  13  s.  41  d.  ? 

4.  What  decimal  of  12°  51'  20"  is  3°  51'  24"? 

5.  What  decimal  of  2  lb.  1  oz.  6  pwt.  is  1  lb.  16  pwt.  ? 

6.  What  decimal  of  4  da.  14  hr.  36  min.  is  9  hr.  5  min.  ? 

7.  What  decimal  of  4  T.  14  cwt.  56  lb.  is  11  cwt.  82  lb.  ? 

8.  What  decimal  of  300  yd.  is  1  fur.  2  rd.  6  yd.  2  ft.  ? 

Comparison  of  Weights. 

78.  The  Troy  pound  and  the  Apothecaries'  pound  are 
equal  in  weight,  and  each  contains  5760  grains ;  the  Avoir- 
dupois pound  contains  7000  grains.     The  grain  is  the  only 


COMPOUND   NUMBERS.  103 

denomination  that  is  the  same  in  all  three  weights ;  hence, 
when  Avoirdupois  Weight  is  to  be  compared  with  either 
Troy  Weight  or  Apothecaries'  Weight,  the  comparison  can 
be  made  by  first  reducing  the  given  denominations  to  grains. 

EXAMPLES. 

1.  What  part  of  a  pound  Avoirdupois  is  a  pound  Troy  ? 

2.  What  part  of  an  ounce  Avoirdupois  is  an  ounce  Troy  ? 

3.  What  part  of  1  3  is  1  dr.  ? 

4.  Express  4  lb.  Avoirdupois  as  the  decimal  of  8  lb.  Troy. 

5.  Find  the  number  of  pennyweights  in  a  pound  Avoir- 
dupois. 

6.  Reduce  5  lb.  6  oz.  Troy  to  Apothecaries'  Weight. 

7.  Reduce  8  lb.  Avoirdupois  to  Apothecaries'  Weight. 

8.  Reduce  12  lb.  Troy  to  Avoirdupois  Weight. 

9.  Reduce  18  lb.  4  oz.  Avoirdupois  to  Troy  Weight. 

10.  A  miner  obtained  $9600  worth  of  gold.  At  $16  an 
ounce  Troy,  what  was  the  weight  of  the  metal  in  Avoirdu- 
pois Weight  ? 

11.  Find  the  value  of  a  silver  cup,  weighing  1  lb.  11  oz. 
Avoirdupois,  at  $1.95  per  ounce  Troy. 

12.  Find  the  amount  gained  by  a  druggist,  who  buys  16 
lb.  Avoirdupois  of  drugs  at  $2.75  per  pound,  and  sells  the 
same  at  20  cts.  per  dram.  Apothecaries'  Weight. 

Comparison  of  Money. 

79.  The  relations  between  United  States  Money  and  the 
moneys  of  other  countries  vary  from  time  to  time.  It  is 
customary  for  the  Secretary  of  the  Treasury  to  publish 
annually  a  table  giving  the  values  of  the  standard  coins  in 
United  States  Money.  The  values  given  Jan-  1st,  1889 
are  as  follows : 


104 


ARITHMETIC. 


Value  in 

COUNTRY. 

Monetary  Unit. 

U.S. 
Money. 

Divisions  of  Units. 

Argentine  Rep. 

Peso    .     .     .     . 

•10.965 

100  centavos  =  1  peso. 

Austria     .     .     . 

Florin      .     .     . 

.336 

100  kreuzers  =^1  florin. 

Belgium    .     .     . 

Franc 

.193 

100  centimes  =  1  franc. 

Bolivia     .     .     . 

Boliviano     .     . 

.68 

100  centavos  =  1  boliviano 

Brazil  .... 

Milreis     .     .     . 

.546 

1000  reis          =  1  milreis. 

British    Posses- 

sions NA. .     . 

Dollar     .     .     . 

LOO 

100  cents        =  1  dollar. 

Chili    .... 

Peso    .     .     .     . 

.912 

100  centavos  =  1  peso. 

Cuba    .... 

Peseta     .     .     . 

.926 

100  centimos  =  1  peseta. 

Denmark .     .     . 

Crown      .     .     . 

.268 

100  ore            =  1  crown. 

Ecuador   .     .     . 

Sucre  .     .     .     . 

.68 

100  centavos  =  1  sucre. 

Egypt  .... 

Pound      .     .     . 

4.943 

100  piastres    —1  pound. 

France      ,     .     . 

Franc      .     .     . 

.193 

100  centimes  =  1  franc. 

German  Empire 

Mark  .     .     .     . 

.238 

100  pfennig    =  1  mark. 

Great  Britain    . 

Pound  Sterling 

4.8665 

20  shillings  =  1  pound. 

Greece      .     .     . 

Drachma      .     . 

.193 

100  lepta        =  1  drachma. 

Guatemala    .     . 

Peso    .     .     .     . 

.68 

100  centavos  =  1  peso. 

Hayti   .... 

Gourde 

.     , 

.965 

100  centavos  =  1  gourde. 

Honduras      .     . 

Peso    . 

, 

.68 

100  centavos  =  1  peso. 

India    .... 

Rupee 

.323 

16  annas       =  1  rupee. 

Italy     .... 

Lira    . 

.     . 

.193 

100centesimit=l  lira. 

Japan  .... 

Y^n  Silver     . 

.997) 

.734/ 

100  sens          =  1  yen. 

Liberia     .     .     . 

Dollar     .     .     . 

1.00 

100  cents        =  1  dollar. 

Mexico     .     .     . 

Dollar 

.739 

100  centavos  =  1  dollar. 

Netherlands .     . 

Florin 

.402 

100  cents        =  1  florin. 

Nicaragua     .     , 

Peso    . 

.68 

100  centavos  =  1  peso. 

Norway     .     .     . 

Crown 

.268 

100  ore           :=  1  crown. 

Peru     .... 

Sol      . 

.68 

100  centavos  =  1  sol. 

Portugal  .     .     . 

Milreis 

LOS 

1000  reis           =  1  milreis. 

Russia .... 

Rouble 

.544 

100  copecks  =  1  rouble. 

Spain   .... 

Peseta 

.193 

100  centimos  =  1  peseta. 

Sweden     .     .     . 

Crown 

.268 

100  ore           =  1  crown. 

Switzerland  .     . 

Franc . 

.193 

100  centimes  =  1  franc. 

Tripoli      .     .     . 

Mahbub 

.614 

20  piastres    =  1  mahbub. 

Turkey     .     .     . 

Piastre 

.044 

40  paras        =  1  piastre. 

U.S.of  Columbia 

Peso    . 

.68 

100  centavos  =  1  peso. 

Venezuela     .     . 

Bolivar 

•     • 

.136 

100  centimos  =  1  bolivar. 

Note.  In  all  answers  in  English  Money  reject  fractions  of  a  farthing 
less  than  one  half,  and  when  the  fraction  of  a  farthing  equals  or  ex- 
ceeds one  half,  reckon  it  as  another  farthing.  In  the  moneys  of  all 
countries  having  a  decimal  system,  treat  like  United  States  Money, 
retaining  only  two  decimal  places  in  the  answer  (for  Brazil  and  Portu- 
gal, three  decimal  places). 


COMPOUND  NUMBERS.  105 

EXAMPLES. 

1.  Eeduce  £S  12  s.  6  d.  .to  United  States  money. 

2.  Reduce  16  guineas  to  United  States  money. 

3.  Eeduce  $128.42  to  English  money. 

4.  Eeduce  172.46  francs  to  United  States  money. 

5.  Eeduce  $45.36  to  French  money. 

6.  Eeduce  64.35  marks  to  United  States  money. 

7.  Eeduce  $75.50  to  German  money. 

8.  Eeduce  32  roubles  to  United  States  money. 

9.  Eeduce  75  crowns  to  United  States  money, 

10.  Eeduce  $114.25  to  Austrian  money. 

11.  Eeduce  £16  15  s.  to  French  money. 

12.  Eeduce  22.25  marks  to  English  money. 

13.  Eeduce  £17  9s.  3d.  to  Federal  money,  taking  4s.  6d. 
=  $1.00. 

14.  If  the  value  of  a  pound  sterling  is  $4.85,  and  of  a 
franc  is  19^  cts.,  what  is  the  equivalent  in  francs  of  2  s.  4  d.? 

Eectangular  Surfaces. 

80.  Any  part  of  a  flat  surface  taken  by  itself  is  called  a 
plane  figure.  The  extent  of  surface  of  a  plane  figure  is 
called  the  area,  and  the  distance  around  it  is  called  the 
perimeter. 

A  rectangle  is  a  plane  figure  having  four  straight  sides 
and  four  right  angles.  When  the  four  sides  of  a  rectangle 
are  all  equal,  it  is  called  a  square. 

A  rectangle  has  two  dimensions  —  length  and  breadth. 


106 


ARITHMETIC. 


I i I I 


The  unit  of  surface  is  a  square,  each  side  of  which  is  a 
unit  of  length.  For  example,  a  square  foot  is  a  square  1  ft, 
long  and  1  ft.  wide. 

Suppose  we  have  a  rectangle' 4  in.  long  and  3  in.  wide. 
Divide  the  length  into  four  equal 
parts,  and  the  width  into  three  equal 
parts,  and  draw  lines  through  the 
points  of  division  as  represented  in 
the  figure.  The  rectangle  is  thus 
divided  into  square  inches.  Upon 
each  inch  of  length  there  is  constructed  a  square  inch,  mak- 
ing a  row  of  4  sq.  in. ;  since  the  rectangle  is  3  in.  wide, 
there  are  3  rows,  each  containing  4  sq.  in.,  making  3  times 
4  sq.  in.,  which  equals  12  sq.  in.  Hence,  to  find  the  area  of 
a  rectangle,  multiply  together  the  length  and  breadth  expressed 
in  the  same  linear  units,  and  the  result  is  the  area  exp)ressed  in 
square  units  of  the  same  name. 

The  quotient  arising  from  dividing  the  product  of  two 
factors  by  one  of  the  factors  is  the  other  factor ;  hence,  if 
the  area  of  a  rectangle  he  divided  by  one  dimension,  the  quo- 


tient is  the  other  dimension. 


I.   Find  the  area  of  a  rectangular  table  whose  length  is 
6  ft.  4  in.  and  width  4  ft.  9  in. 


6  ft.  4  in. 
12 
72 
_4 
76  in. 


4  ft.  9  in. 
12 
48 

9 
§7  in. 


76 
_57 
532 
380 
144) 4332 (30  sq.ft. 
432 
12  sq.  in. 


A71S.  30  sq.  ft.  12  sq.  in. 

The  two  dimensions  must  be  reduced  to  inches,  and  their  product 
denotes  the  number  of  square  inches.  Since  the  dimensions  are  given 
in  feet  and  inches,  the  area  should  be  expressed  in  square  feet  and 
square  inches. 


COMPOUND   NUMBERS.  107 

II.   The  area  of  a  rectangular  floor  is  224  sq.  ft.,  and  its 
length  is  16  ft. ;  find  its  width. 

1fi^224.  "^^^  number  of  square  feet  in  the  area  divided  by 

..  „  the  number  of  feet  in  length   equals  the  number  of 

^^  "•         feet  in  breadth. 

EXAMPLES. 

1.  How  many  square  yards  are  there  in  a  floor  24  ft. 
long  and  14  ft.  wide  ? 

2.  Find  the  cost  of  oil-cloth  to  cover  a  floor  15  ft.  long 
and  10^  ft.  wide  at  45  cts.  per  square  yard. 

3.  If  a  floor  contains  35  sq.  yd.,  and  is  21  ft.  long,  what 
is  its  width  ? 

4.  How  many  blocks  1  ft.  square  will  it  take  to  pave  an 
alley  54  rd.  long  and  8  ft.  wide  ? 

5.  How  many  acres  are  there  in  a  field  72  rd.  long  and 
60  rd.  wide  ? 

6.  Find  the  area  of  a  square  field  each  of  whose  dimen- 
sions is  65  rd. 

7.  Find  the  value  of  a  field  180  rd.  long  and  94-J-  rd.  wide 
at  $18  an  acre. 

8.  The  area  of  a  field  is  10  A.,  and  its  width  is  20  rd. ; 
what  is  its  length  ? 

9.  Find  the  cost  of  paving  a  street  1028  ft.  long  and 
63-J-  ft.  wide  at  $3.25  a  square  yard. 

10.  What  length  of  road  38^  ft.  wide  will  contain  3  A.  ? 

11.  A  field  is  38-^  rd.  long   and  37i  rd.  wide ;   find  its 
area  in  acres  and  square  rods. 

12.  A  path  is  26  ft.  8  in.  long  and  5  ft.  3  in.  wide  j  find 
its  area  in  square  feet. 


108  ARITHMETIC. 

13.  A  garden,  containing  |  of  an  acre,  measures  198  ft.  on 
one  sid5 ;  find  the  length  of  the  other  side. 

14.  How  many  acres  are  there  in  a  field  15.72  ch.  long 
and  8.95  ch.  wide  ? 

15.  A  building  lot,  containing  |-  of  an  acre,  has  a  frontage 
of  90  ft. ;  how  far  back  does  it  extend  ? 

16.  Find  the  difference  in  area  between  two  lots  of  land, 
one  of  which  is  30  rd.  square,  and  the  other  contains  30 
sq.  rd. 

17.  A  field,  containing  14  A.,  is  56  rd.  long ;  what  is  its 
width  ?  Find  the  cost  of  building  a  fence  around  it  at  45 
cts.  a  rod. 

18.  Find  the  cost  of  slating  a  roof  40  ft.  long  and  each 
of  the  two  sides  20  ft.  wide,  at  ^10  per  square  of  100  sq.  ft. 

19.  At  $37.50  per  acre,  find  the  cost  of  a  field  55.33  ch 
long  and  148  rd.  3  yd.  1  ft.  6  in.  wide. 

20.  How  many  bricks  7^  in.  long  and  3^  in.  wide  will  it 
take  to  lay  a  walk  462  ft.  long  and  6^  ft.  wide  ? 

21.  How  many  tiles  9  in.  square  will  it  take  to  pave  a 
court  114  ft.  long  and  48  ft.  wide  ? 

22.  How  many  boards,  each  14  ft.  long  and  7^  in.  wide, 
will  it  take  to  build  a  platform  42  yd.  long  and  30  yd.  wide  ? 

23.  The  area  of  a  field  is  49  sq.  rd.  22  sq.  yd.  6  sq.  ft.  108 
sq.  in.,  and  the  length  is  7  rd.  4  yd.  1  ft.  6  in. ;  find  the 
width. 

24.  Find  the  cost,  at  60  cents  a  square  yard,  of  making 
a  gravel  path  5  ft.  wide  around  a  garden  78  ft.  long  and  42 
ft.  wide : 

(i)    when  the  path  is  outside  the  garden, 
(ii)    when  the  given  dimensions  include  both  garden  and  path. 


COMPOUND  NUMBERS. 


109 


JIectangular  Volumes. 

81.  A  rectangular  parallelepiped  is  a  volume  bounded  by 

six  rectangular  surfaces.  The  bounding  surfaces  are  called 
faces,  and  the  bounding  lines  are  called  edges.  The  faxies 
taken  together  constitute  the  surface,  and  the  lower  face  is 
called  the  base.  When  the  faces  are  six  equal  squares,  the 
volume  is  called  a  cube. 

A  rectangular  parallelopiped  has  three  dimensions  — 
length,  breadth,  and  thickness. 

The  unit  of  volume  is  a  cube,  each  dimension  of  which  is 
a  unit  of  length.  For  example,  a  cubic  foot  is  a  cube  1  ft. 
long,  1  ft.  wide,  and  1  ft.  thick. 

Suppose  we  have  a  rectangular  parallelopiped  4  in.  long, 
3  in.  wide,  and  2  in.  thick.  The  upper  face  contains  3  times 
i  sq.  in.,  or  12  sq.  in. ;  and  if 
fche  parall-elopiped  were  1  in. 
thick,  there  would  be  as 
many  cubic  inches  as  there 
are  square  inches  on  the 
upper  face.  But  the  paral- 
lelopiped is  2  in.  thick,  and 
must,  therefore,  contain  twice 
as  many  cubic  inches  as  it 

would  if  it  were  only  1  in.  thick,  or  2  times  3  times  4  cu.  in., 
which  equals  24  cu.  in.  Hence,  to  find  the  cubic  contents  of 
a  rectangular  parallelopiped,  multiply  together  the  three  dimen- 
sions expressed  in  the  same  linear  units,  and  the  result  is  the 
cubic  contends  expressed  in  cubic  units  of  the  same  name. 

The  quotient  arising  from  dividing  the  product  of  three 
factors  by  the  product  of  two  of  the  factors  is  the  third 
factor ;  hence,  if  the  cubic  contents  of  a  rectarigular  parallelo- 
piped be  divided  by  the  product  of  two  dimensions,  the  quotient 
is  the  third  dimension. 


llrik  \  ^  \  ^ 

\ 

Ik  \  \  ^-^- 

Bi  ^  ^"  ^  "^ 

■  ■':: 

iilltll 

IIMItilil 

ill 

no 


ARITHMETIC. 


I.   Find  the  number  of  gallons  in  a  cistern  7  ft.  long,  6  ft. 
wide,  and  5  ft.  deep.  » 

7  X  6  X  5  =  210  cu.  ft. 

1728 

210 
17280 
3456 
231)362880  cu.  in.(1570if  gal. 
231 
1318 
1155 


1638 
1617 


The  product  of  the  three 
dimensions  gives  210cu.  ft.  as 
the  cubic  contents,  and  this 
equals  362880  cu.  in.  Since 
there  are  2.31  cu.  in.  in  1  gal., 
there  are  as  many  gallons  in 
362880  cu.  in.  as  231  is  con- 
tained times  in  362880,  which 
equals  1570}^  gal. 


210 
231 


10 
11 


II.   A  rectangular  solid,  whose  cubic  contents  are  924  cu. 
ft.,  is  22  ft.  long  and  7  ft.  wide ;  what  is  its  thickness  ? 

22  X  7  =  154  The  product  of  the   two  known  dimensions  is 

154)924(6  ft.         154,  and  dividing  924,  the  cubic  contents,  by  154 


924 


gives  6,  the  number  of  feet  in  thickness. 


EXAMPLES. 

1.  Find  the  number  of  cubic  feet  of  air  in  a  room  24  ft. 
long,  18  ft.  wide,  and  10^  ft.  high. 

2.  Find  the  cost,  at  30  cts.  a  cubic  yard,  of  digging  a 
cellar  56  ft.  long,  28  ft.  wide,  and  9  ft.  deep. 

3.  A  reservoir  30  ft.  wide  and  12  ft.  deep  contains  960 

cu.  yd. ;  what  is  its  length  ? 

4.  A  vat   12  ft.  square    contains  1368  cu.  ft. ;    find  its 
depth. 

5.  Find  the  volume  of  a  cube  whose  edge  measures  2  ft. 
9  in. 


COMPOUND   NUMBERS.  Ill 

6.  How  many  cubic  feet  are  there  in  a  stick  (n  timber 
17  ft.  long,  15  in.  wide,  and  8  in.  thick  ? 

7.  What  must  be  the  length  of  a  stick  of  timber,  1^  ft. 
square  at  the  end,  to  contain  100  cu.  ft.  ? 

8.  If  a  box  5  ft.  4  in.  high  contains  36  cu.  ft.,  what  is 
the  area  of  the  base  ? 

9.  How  many  cords  of  wood  are  there  in  a  pile  36  ft. 
long,  6  ft.  high,  and  4  ft.  wide  ? 

10.  Find  the  value  of  a  pile  of  wood  28  ft.  ?.ong,  5^  ft. 
high,  and  4  ft.  wide,  at  $3.25  a  cord. 

11.  What  must  be  the  length  of  a  pile  of  wood,  4Jft. 
high  and  3^  ft.  wide,  to  contain  2  cords  ? 

12.  If  a  cubic  foot  of  ice  weighs  58.1  lb.,  how  many  tons 
will  be  contained  in  an  ice-house  45  ft.  long,  32  ft.  wide,  and 
20  ft.  high  ? 

13.  Find  the  number  of  gallons  in  a  tank  3  ft.  6  in.  long, 
2  ft.  4  in.  wide,  and  1  ft.  10  in.  deep. 

14.  Find  the  number  of  gallons  in  a  cistern  5^  ft.  square 
and  7  ft.  deep.  ' 

15.  Find  the  value,  at  90  cts.  a  bushel,  of  the  grain  that 
will  be  contained  in  a  bin  14  ft.  long,  10  ft.  wide,  and  5  ft. 
deep. 

16.  Find  the  depth  of  a  bin  necessary  to  hold  160  bu.,  if 
its  length  is  9  ft.  and  its  width  5  ft. 

17.  How  many  bricks  will  it  take  to  build  a  wall  56  ft. 
long,  9  ft.  high,  and  4  ft.  thick,  each  brick  being  8  in.  long, 
4|-  in.  wide,  and  2\  in.  thick  ? 

18.  How  many  stones,  10  in.  long,  9  in.  broad,  and  4  in. 
thick,  would  it  require  to  build  a  wall  80  ft.  long,  20  ft. 
high,  and  2^  ft.  thick  ? 


112  ARITHMETIC. 

19.  How  many  square  feet  are  there  on  the  surface  of  a 
box  2^  ft.  long,  2  ft.  wide,  and  3  ft.  deep  ? 

20.  How  many  square  feet  are  there  on  the  surface  of  a 
cubical  box,  each  of  whose  dimensions  is  2|  ft.  ? 

21.  A  river,  30  ft.  deep  and  20  yd.  wide,  flows  4  mi.  an 
hour.  Find  the  number  of  cubic  feet  of  water  which  pass 
a  given  point  in  a  minute. 

22.  How  many  cords  of  stone  will  it  take  to  build  a  wall, 
2  ft.  thick  and  6  ft.  high,  about  a  cellar  whose  interior  di- 
mensions, when  the  wall  is  completed,  shall  be  20  ft.  long 
and  16  ft.  wide  ? 

MISCELLANEOUS  EXAMPLES. 

1.  What  is  the  cost  of  5  T.  7  cwt.  24  lb.  of  hay  at  $16 
per  ton  ? 

2.  Find  the  cost  of  1  qt.  of  olive  oil  when  1  doz.  pints 
cost  $3.45. 

3.  What  is  the  value  of  1  doz.  silver  spoons,  each  weigh- 
ing 2  oz.  16  pwt.  16  gr.,  at  $1.15  per  ounce  ? 

4.  How  many  times  will  a  wheel,  9  ft.  4  in.  in  circum- 
ference, turn  in  crossing  a  bridge,  the  length  of  which  is 
54  rd.  4  yd.  2  ft.  4in.  ? 

5.  Reduce  44920.9025  hr.  to  years  (of  365  days),  days, 
hours,  minutes,  and  seconds. 

6.  When  coal  is  worth  $6.25  a  long  ton,  what  is  the 
expense  of  a  coal  fire  for  the  month  of  January,  allowing 
351b.  a  day? 

7.  If  one  man  performs  a  piece  of  labor  in  2  da.  13  hr. 
41  min.,  how  long  would  it  take  10  men  to  perform  the 
same  work  ? 


COMPOUND  NUMBERS.  113 

8.  A  rectangular  field  measures  30  rd.  6  ft.  by  21  rd.  11 
ft. ;  find  the  area  in  acres,  square  rods,  and  square  feet. 

9.  What  is  the  value  of  a  piece  of  ground,  16^  rd.  long 
and  27^  yd.  wide,  at  1  s.  4  d.  per  square  foot  ? 

10.  If  12^  yd.  of  silk  that  is  f  yd.  wide  will  make  a  dress, 
how  many  yards  of  muslin  that  is  If  yd.  wide  will  be  re- 
quired to  line  it  ? 

11.  A  cellar  is  to  be  dug  30  ft.  long  and  20  ft.  wide ;  at 
what  depth  will  50  cu.  yd.  of  .earth  have  been  removed  ? 

12.  It  takes  8  hr.  40  min.  to  fiU  a  certain  cistern ;  what 
part  of  it  has  been  filled  after  water  has  been  running  in 
2  hr.  45  min.  45  sec.  ? 

13.  How  long  will  it  take  a  man  to  walk  48  mi.  210  rd. 
12  ft.,  if  he  walks  15  mi.  in  4  hr.  15  min.  ? 

14.  How  many  silver  dollars,  each  weighing  412^  gr.,  can 
be  coined  from  a  bar  of  silver  weighing  8|-  lb.  Avoirdupois  ? 

15.  A  man  earns  f  325  in  2^  months,  and  spends  in  6 
months  what  he  earns  in  4^  months  ;  what  does  he  save  in 
a  year  ? 

16.  A  regiment  of  troops  enlisted  for  9  months  and  was 
discharged  May  25th,  1863,  which  was  1  mo.  12  da.  after 
the  term  of  service  had  expired.  Find  the-  date  when  they 
enlisted. 

17.  A  cable  that  weighs  one  ton  per  mile  weighs  how 
much  per  foot  ? 

18.  The  velocity  of  a  body  is  40  mi.  per  hour;  what  is  it 
expressed  in  feet  per  second  ? 

19.  If  a  train  travels  40  ft.  in  a  second,  how  far  will  it 
travel  in  1  hr.  31  min.  18  sec.  ? 

20.  Divide  2  gal.  1  qt.  1.02  pt.  by  17.  Express  the  result 
in  pints ;  also  in  the  decimal  of  a  barrel. 


114  AKITHMETIC. 

21.  Find  ^  of  3  mi.  2  fur.  25  rd.  3  yd.  2  ft.  6  in.  as  a  com- 
pound number;  reduce  it  to  chains  and  the  decimal  of  a 
chain. 

22.  Express  as  a  fraction  of  an  acre  the  ground  taken  up 
by  a  path  3  ft.  broad  round  a  house,  the  front  of  which  is 
57  ft.,  and  side  is  37  ft. 

23.  If  2  cu.  in.  of  iron  weigh  as  much  as  15  ri\x.  in.  of 
water,  and  1  cu.  ft.  of  water  weighs  1000  oz.,  find  the  weight 
in  tons  of  1  cu.  yd.  of  iron. 

24.  If  a  grocer's  scales  give  only  15  oz.  4  dr.  for  a  pound, 
out  of  how  much  money  is  a  customer  cheated  who  buys 
sugar  to  the  amount  of  $55.04  ? 

25.  If  2  A.  3E.  4  P.  be  multiplied  by  2f,  what  part  is 
the  product  of  15  A.  1  R.  2  P.  ? 

26.  A  owns  -j^  of  a  field,  and  B  owns  the  remainder;  | 
of  the  difference  between  their  shares  is  5  A.  3R.  16^  ¥. 
What  is  B's  share  in  acres  ? 

27.  Thirty-six  persons  buy  2766  A.  3  R.  12  P.  of  land  on 
equal  shares.  What  does  one  man  receive,  who  sells  f  of 
his  share  at  Is.  9  d.  2  far.  per  square  rod?  [Give  the 
answer  in  pounds  and  the  decimal  of  a  pound.] 

28.  Pind  the  weight  of  500000  bricks  at  4  lb.  2  oz.  each, 
and  the  cost  in  dollars  and  cents,  at  27  s.  6  d.  a  thousand, 
allowing  4  s.  2  d.  to  make  a  dollar. 

29.  Reduce  12  T.  8  cwt.  551b.  3  oz.  3|dr.  Avoirdupois 
Weight  to  pounds  and  the  decimal  of  a  pound ;  then  reduce 
to  Troy  Weight. 

30.  If  a  body  revolves  uniformly  in  the  circumference  of 
a  circle  at  the  rate  of  12°  15'  25"  per  minute,  how  long  is  it 
in  performing  a  complete  revolution  ? 


THE  METRIC   SYSTEM.  115 


CHAPTER  VI. 

THE  METRIC  SYSTEM. 

82.  The  metric  system  is  a  system  of  weights  and  meas- 
ures based  upon  the  decimal  system  of  notation.  It  has 
been  adopted  by  nearly  all  civilized  nations,  and  its  use  has 
been  legalized  in  the  United  States  and  Great  Britain. 

The  unit  of  length  is  the  meter,  and  from  it  are  derived 
the  units  of  surface,  volume,  capacity,  and  weight.  The 
length  of  the  meter  is  defined  by  a  bar  kept  at  Paris.  This 
length  was  adopted  in  1799,  and  is  one  ten-millionth  of  the 
distance  from  the  equator  to  either  pole,  as  calculated  at 
that  time.  However,  later  calculations  have  proved  that 
the  meter  is  a  very  small  fraction  shorter  than  one  ten- 
millionth  of  this  distance  on  the  earth's  surface. 

From  the  different  units  are  derived  other  denominations 
by  adding  prefixes.  The  prefixes  for  the  fractional  parts  of 
the  unit  are  derived  from  Latin  numerals,  and  those  for  the 
multiples  of  the  unit  are  derived  from  Greek  numerals. 


Deci  means  tenth. 
Centi     "      hundredth. 
Milli      "     thousandth. 


Deka  means  ten. 
Hgkto      "      hundred. 
Kilo         "      thousand. 
Myria      "      ten  thousand. 


In  the  following  tables  the  denominations  in  common  use 
are  printed  in  full-faced  type.  Abbreviations  beginning 
with  a  small  letter  denote  a  fractional  part  of  the  principal 
unit ;  abbreviations  beginning  with  a  capital  letter  denote 
a  multiple  of  the  unit. 


116  ARITHMETIC. 

Measures  of  Length. 

83.  TABLE. 

10  millimeters  C""")  =  1  centimeter  (•""). 

10  centimeters  =  1  decimeter  (^'"). 

10  decimeters  =  1  meter  ('"). 

10  meters  =  1  dekameter  (^"'). 

10  dekameters  =  1  hektometer  (^'"). 

10  hektometers  =  1  kilometer  (^'"). 

10  kilometers  =  1  myriameter  {^^), 

Measures  of  Surface. 

84.  The  units  of  surface  are  squares  whose  dimensions 
are  the  corresponding  linear  units ;  hence  it  takes  10  times 
10,  or  100,  of  one  denomination  to  make  one  of  the  next 
higher.  For  measuring  small  surfaces  the  principal  unit  is 
the  square  meter. 

TABLE. 

100  square  millimeters  (««"»'")  =  1  square  centimeter  (*'i*™). 
100  square  centimeters  =  1  square  decimeter  (sq^"'). 

100  square  decimeters  =  1  square  meter  ("'i'"). 

100  square  meters  =  1  square  dekameter  (^i^m), 

100  square  dekameters  =  1  square  hektometer  (^^Hm^. 

100  square  hektometers  =  1  square  kilometer  (^q  ^m^ 

In  measuring  land  the  square  meter  is  called  a  centar 
(^),  the  square  dekameter  is  called  an  ar  (*),  and  the  square 
hektometer  is  called  a  hektar  (°*). 

Measures  of  Volume. 

85.  The  units  of  volume  are  cubes  whose  dimensions  are 
the  corresponding  linear  units ;  hence  it  takes  10  times  10 


THE   METRIC   SYSTEM.  1x7 

times  10,  or  1000,  of  one  denomination  to  make  one  of  the 
next  higher.     The  principal  unit  is  the  cubic  meter. 

TABLE. 

1000  cubic  millimeters  (<="'"'")  =  1  cubic  centimeter  («="«^«). 
1000  cubic  centimeters  =  1  cubic  decimeter  (•=«<!">). 

1000  cubic  decimeters  =  1  cubic  meter  C^™). 

Ifi  measuring  wood  the  cubic  meter  is  called  a  ster  (**) ; 
one  tenth  of  a  cubic  meter  is  a  decister  (*^),  and  ten  cubic 
meters  are  a  dekaster  (^^). 

Measures  of  Capacity. 

86.  The  unit  of  capacity  is  a  liter,  which  equals  a  cubic 
decimeter. 

TABLE. 

10  milliliters  C"^)  =  1  centiliter  (<=0. 
10  centiliters  =  1  deciliter  (<"). 
10  deciliters  =  1  liter  ('). 

10  liters  =  1  dekaliter  (»'). 

10  dekaliters         =  1  hektoliter  (™). 
10  hektoliters        =  1  kiloliter  (k>). 


Weight. 

87.  The  unit  of  weight  is  a  gram,  which  equals  the 
weight  of  a  cubic  centimeter  of  water  at  its  greatest 
density. 

TABLE. 

10  milligrams  (™s)  =  1  centigram  (•«). 
10  centigrams  =  1  decigram  {^^). 

10  decigrams  =  1  gram  (»). 


118  ARITHMETIC. 

10  grams  =  1  dekagram  (^«). 

10  dekagrams    =  1  hektogram  (°8). 

10  hektograms  =  1  kilogram  (^«)  or  kilo  (^). 

10  kilograms     =  1  myriagram  (^^). 

10  myriagrams  =  1  quintal  {^). 

10  quintals        =  1  tonneau  or  ton  (^). 

A  cubic  centimeter  or  milliliter  of  water  weighs  a  gram. 
A  cubic  decimeter  or  liter  of  water  weighs  a  kilogram. 
A  cubic  meter  or  kiloliter  of  water  weighs  a  ton. 

88.  A  metric  number  can  be  reduced  to  another  denomi- 
nation by  simply  moving  the  decimal  point.  Por  example, 
1945.2^=  1.9452^*^,  because  dividing  by  10  three  times  is  the 
same  as  moving  the  decimal  point  three  places  to  the  left ; 
3.726""=  372.6*=™,  because  multiplying  by  10  twice  is  the 
same  as  moving  the  decimal  point  two  places  to  the  right. 

In  ]:educing  from  a  lower  to  a  higher  denomination,  move 
the  decimal  point  to  the  left  as  many  places  as  there  are  in- 
tervals in  the  table  between  the  given  denomination  and  the 
required  denomination. 

In  reducing  from  a  higher  to  a  lower  denomination,  move 
the  decimal  point  to  the  right  as  many  places  as  there  are  in- 
tervals in  the  table  between  the  given  denomination  and  the 
required  denomination. 

In  measures  of  surface  it  takes  100  of  one  denomination  to 
make  one  of  the  next  higher ;  hence  the  decimal  point  must 
be  moved  two  places  for  every  interval. 

In  measures  of  volume  it  takes  1000  of  one  denomination 
to  make  one  of  the  next  higher ;  hence  the  decimal  point 
must  be  moved  three  places  for  every  interval. 

EXAMPLES. 

1.  Reduce  6453™  to  kilometers. 

2.  Beduce  4.15™  to  centimeters. 


THE   METRIC   SYSTEM.  119 

3.  Reduce  6.45^  to  milliliters. 

4.  In  9780*"  how  many  kilometers  ? 

5.  Write  4^"^,  5"%  2"",  and  S"""  as  meters. 

6.  Write  84°^,  8^,  and  92*=^  as  liters. 

7.  Write  1872.6'"  as  kilometers ;  as  centimeters ;  as  mil- 
limeters. 

8.  Write  67.43^*^  as  tons  ;  as  grams  ;  as  milligrams. 

9.  Write  7529*=^  as  liters ;  as  hektoliters. 

10.  Write  96547™^  as  grams ;  as  kilograms. 

11.  W^ite  7.653^^  as  liters ;  as  centiliters. 

12.  43720"""  equals  how  many  meters  ?    how  many  centi- 
neters  ?   what  fraction  of  a  kilometer  ? 

13.  Write  S^%  6%  and  72*=»  as  hektars. 

14.  Write  968.32"*!  m  ^g  ^^^.g .  g^g  square  centimeters. 

15.  Write  546.31^  as  square  kilometers  ;  as  centars. 

16.  Write   8915200''^*=™   as   square   meters;    as   ars;    as 
hektars. 

17.  How  many  cubic  millimeters  are  there  in  a  cubic 
dekameter  ? 

18.  In  2.15*^""'  how  many  cubic  millimeters  ? 

19.  Express  2328000*="*^™  as  sters;  as  dekasters;  as  deci- 
sters. 

20.  What  is  the  value  in  cubic  centimeters  of  297^  ?   in 
cubic  meters  ?    in  cubic  kilometers  ? 

21.  Write  0.853*^""  as  hektoliters;    as  liters;    as  centi- 
liters ;  as  cubic  centimeters. 

22.  Write  81470*^"*=™  as  liters;    as  hektoliters;   as  cubic 
meters. 


120  ARITHMETIC. 

23.  Express  29.73^^  as  liters  ;  as  centiliters ;  as  cubic 
meters ;  as  cubic  centimeters. 

24.  What  is  the  weight  of  276.5'="  «=™  of  water? 

25.  Find  the  weight  in  kilograms  of  0.0316'="'"  of  water. 

26.  How  many  decigrams  does  a  dekaliter  of  watei 
weigh  ? 

27.  What  is  the  weight  in  kilograms  of  12"'  of  water  ? 

28.  How  much  will  a  cubic  hektometer  of  water  weigh 
in  kilograms  ?  Express  the  same  quantity  of  water  in 
liters. 

29.  What  is  the  amount  of  34789.56^  of  water  in  cubic 
centimeters  ?  in  cubic  meters  ?  in  cubic  kilometers  ?  its 
weight  in  grams  ?   in  kilograms  ? 

30.  What  is  the  amount  of  294.7361™  of  water  in  cubic 
meters  ?  in  liters  ?  in  cubic  millimeters  ?  its  weight  in 
tons  ?   in  grams  ?   in  milligrams  ? 

89.  All  operations  in  the  metric  system  are  performed 
as  in  decimal  fractions.  If  metric  numbers  are  expressed 
in  different  denominations,  they  must  be  reduced  to  the 
same  denomination  before  they  can  be  added  or  subtracted. 

EXAMPLES. 

1.  How  many  grams  are  there  in  23.45^*^  and  15.8°^  ? 

2.  Add-  together  1.23^,  306.7""",  0.5219^™,  and  2.91",  and 

express  the  sum  in  centimeters. 

3.  Express  the  sum  of  305"^^,  218°^,  and  7"^  in  kilograms. 

4.  Express  in  square  meters  l^-^^  250*-  ISO"**-  1500'"i ''™. 

5.  Eind  the  sum  of  1871'="'=™,  541',  and  4.51™,  and  give 
the  answer  in  liters. 


THE  METRIC   SYSTEM.  121 

6.  Express  in  cubic  meters  7^"^"+  54^'+  .03°'+  5400'="'*'". 

7.  Multiply  17.28«  by  312500,  and  give  the  product  in 
kilos. 

8.  Multiply  the  sum  of  7^,  823",  and  125"^  by  5.12. 

9.  Divide  3035.25*""  by  0.0375. 

10.  Divide  2700^1  by  90«=l 

11.  Find  the  value  in  cubic  decimeters  of  \^  of  87*^"* 

gOcudm  QQQcucm 

12.  What  will  100'  of  mercury  weigli,  mercury  being  13.5 
times  as  heavy  as  water  ? 

13.  What  weight  of  mercury  will  a  vessel  contain  whose 
capacity  is  20*="<='"? 

14.  What  is  the  weight  of  water  in  a  tank  if  it  would 
take  98  minutes  to  empty  it  at  the  rate  of  8.7'  a  minute  ? 
If  it  were  filled  with  oil  at  f  18.75  a  hektoliter,  what  would 
the  contents  be  worth  ? 

Rectangular  Surfaces  and  Volumes. 
90.    For  principles  see  sections  80  and  81. 

EXAMPLES. 

1.  How  many  square  decimeters  are  there  in  a  board 
4*"  long  and  0.4"*  wide  ? 

2.  How  many  hektars  are  there  in  a  strip  of  land  62*'"' 
broa^  and  1.7^"  long  ? 

3.  How  many  centars   are  there  in  a  sidewalk  0.42^™ 
long  and  2.8"  wide  ? 

4.  How  many  ars  are  there  in  a  field  54"  long  and  28.4™ 
wide?  / : 


122  ARITHMETIC. 

5.  Two  rectangular  fields  are  G^'"  long  and  IQ^*"  wide 
and  7^™  long  and  IS^""  wide  respectively ;  how  many  more 
hektars  are  there  in  the  second  than  in  the  first  ? 

6.  How  many  bricks,  each  20°™  long  and  lO^""  wide,  will 
it  take  to  pave  a  sidewalk  3.3™  wide  and  1.7^'™  long  ? 

7.  If  a  person  steps  0.8™  at  each  step,  how  many  steps 
will  he  take  in  walking  around  a  rectangular  field,  which 
contains  lOSO^"^,  and  whose  breadth  is  1800™  ? 

8.  A  rectangular  piece  of  ground  is  32™  7*^™  long  and 
19™  S*'"  broad.  Find  the  cost  of  enclosing  it  with  a  path 
1™  5*^™  broad  at  3  francs  5  centimes  a  square  meter : 

(i)  when  the  path  is  outside  the  ground, 
(ii)  when  the  path  is  part  of  the  ground. 

9.  How  many  liters  are  there  in  a  vat  2.8™  long,  2™ 
wide,  and  5*^™  deep  ? 

10.  How  many  liters  are  there  in  a  box  1.2™  long,  8°™ 
wide,  and  50™™  deep  ? 

11.  A  cistern  is  4™  long,  24^™  wide,  and  80*=™  deep.     How 
much  water  will  it  hold  in  cubic  meters  ?   in  liters  ? 

'  12.  How  many  cubic  meters  of  air  will  a  room  contain 
whose  length  is  5™  2^™,  whose  breadth  is  4™,  and  whose 
height  is  35*^™  ?     What  is  the  amount  in  liters  ? 

13.  There  is  a  bin  7.6™  long,  4.3™  wide,  and  3.86™  deep. 
How  many  hektoliters  of  wheat  will  it  contain  ? 

14.  How  many  hektoliters  of  oats  can  be  put  into  a  bin 
that  is  2™  long,  1.3™  wide,  and  1.5™  deep  ? 

15.  What  is  the  cost  of  a  pile  of  wood  whose  dimensions 
are  2™,  1.9™,  and  42.5™,  at  f  2  per  ster  ? 

16.  What  is  the  cost  of  digging  a  cellar  3°'"  wide,  5°™  4™ 
long,  and  2™  6*^  deep,  at  the  rate  of  50  cts.  a  ster  ? 


THE   METRIC    SYSTEM.  123 

17.  How  many  sters  are  there  in  a  wall  24"*  long,  8"*  5**" 
high,  and  52*='"  thick  ?  What  would  be  the  cost  of  building 
it  at  $4.25  a  cubic  meter  ? 

18.  What  weight  of  water  (in  kilograms)  may  be  con- 
tained in  a  cistern  1.75"'  long,  l.S™  broad,  and  0.8'"  deep  ? 

19.  How  many  liters  of  water  can  be  contained  in  a  cis- 
tern 5"™  long,  3""  wide,  and  2*"  deep  ?  What  would  be  the 
weight  of  the  water  in  kilograms  ? 

20.  A  bin  is  3.4™  long,  1.36'"  wide,  and  0.84"  deep.  How 
many  kilograms  of  water  will  it  hold  ?  How  many  hekto- 
1  iters  of  wheat  will  it  contain  ? 

21.  How  many  hektoliters  will  a  bin  hold  that  is  3™  long, 
22*'"  wide,  and  0.015"'"  deep?  How  many  kilograms  of 
water  will  it  hold  ? 

22.  Required  the  weight  in  centigrams  of  the  water  in  a 
vessel  1'"  2*='"  long,  6*^"*  broad,  and  5*"  1'""*  deep. 

23.  A  cistern  is  5""  long,  36'*'"  wide,  and  OO*^'"  deep.  How 
much  water  will  it  hold  in  cubic  meters  ?  in  liters  ?  in 
cubic  centimeters  ?   in  grams  ?   in  kilograms  ? 

24.  A  box  2.3™  long,  196.7'='"  broad,  and  901.9*"™  deep  con- 
tains how  many  liters  ?  If  filled  with  water,  how  many 
grams  would  the  water  weigh  ? 

25.  A  vat  is  6.3™  long,  3™  wide,  and  4.2™  deep.  How  long 
will  it  take  a  water-pipe  to  fill  the  vat,  if  the  current  flows 
at  the  rate  of  3.6^^  a  minute  ? 

26.  A  cubical  cistern  is  6"**  in  each  dimension.  If  1.725^  of 
water  can  flow  out  per  minute,  how  much  must  flow  in  per 
minute  to  fill  it  in  an  hour  ? 

27.  How  many  grams  of  a  liquid  li  times  as  heavy  as 
water  will  fill  a  cube  whose  edge  is  20*=™?  How  many 
liters  ? 


124  ARITHMETIC. 

28.  Find  the  weight  in  grams  of  a  bar  of  gold  l*^*"  long, 
2-i.'^"»  wide,  and  2'^'"  thick,  assuming  the  bar  to  be  19  times  as 
heavy  as  its  own  volume  of  water. 

29.  What  must  be  the  length  of  a  pile  of  wood,  2'"  high 
and  1.25'"  wide,  to  contain  12**  ? 

30.  What  must  be  the  length  of  a  box,  1™  wide  and  1™ 
deep,  to  contain  4500^  ? 

31.  A  cistern  holding  10i-'=""' is  25*^™  wide  and  S'^long; 
find  its  depth  in  centimeters. 

32.  A  roof  10.5™  long  by  5.4™  wide  drains  into  a  tank  1" 
deep  with  a  base  1.25'"  by  2.5"'.  What  dej)th  of  water  must 
fall  on  the  roof  to  fill  the  tank  ? 

33.  How  many  hektars  of  land  can  be  flooded  to  the 
depth  of  5"'"'  from  a  tank  holding  1000 '^  of  water? 

The  Metric  System  Compared  with  the  Common 

System. 

91.  The  following  table  of  equivalents  should  be  com- 
mitted to  memory,  since  by  its  use  weights  and  measures  of 
one  system  can  readily  be  converted  into  weights  and  meas- 
ures of  the  other. 

1  meter  =  39.37  inches. 

1  kilometer       =    0.62138  mile. 

1  square  meter  =  1550  square  inches. 

1  hektar  =    2.471  acres. 

1  cubic  meter    =    1.308  cubic  yards. 

1  ster  =    0.2759  cord. 

1  liter  =    1.0567  liquid  quarts, 

=    0.908  dry  quart. 
1  gram  =  15.432  grains. 

1  kilogram        =    2.2046  pounds  Avoirdupois. 


THE   METRIC    SYSTEM.  126 

I.  Reduce  34*  to  square  rods. 

2.471  A. 

0.34  34a  must  first  be  ted  need  to  hektars,  the 

9884  denomination  given   in   the   equivalent.     Since 

7413  !"«=  2.471  A.,  0.34"a  equals  0.34  of  2.471  A., 

0  84014  A  which   equals    0.84014  A.      Then    reducing    to 

■1  gA  square  rods,  we  have  134.4224  sq.  rd. 

134.42240  sq.  rd. 

II.  Reduce  5  mi.  3  fur.  10  rd.  to  kilometers. 

40)10.00  rd.  0.62138)5.40625(8.7004'^ 

8)3:25  fur.  497104 

5.40625  mi.  435210 

434966 

244000 

The  compound  number  must  first  be  reduced  to  miles,  the  denomi- 
nation given  in  the  equivalent.  Since  lKm  =  0.62138  mi.,  in  5.40625  mi. 
there  are  as  many  kilometers  as  0.62138  is  contained  times  in  6.40625, 
which  equals  8.7004Km. 

In  reducing  from  metric  to  common  weights  and  measures, 
multiply  the  jiumher,  expressed  in  the  denomination  of  the  equiv- 
alent, by  the  equivalent. 

In  reducing  from  common  to  metric  weights  and  measures, 
divide  the  number^  expressed  in  the  denomination  of  the  equiv- 
alent^ by  the  equivalent. 

EXAMPLES. 

1.  Reduce  600^'"  to  miles. 

2.  Reduce  10°^  to  feet. 

3.  Reduce  42.5^  to  gallons. 

4.  Reduce  16. 75H»  to  bushels. 

5.  Reduce  18''  to  cords. 

6.  Reduce  50^  to  grains. 


126  ARITHMETIC. 

7.  Eeduce  126*  to  ounces  Avoirdupois. 

8.  Eeduce  20^*^  to  pounds  Avoirdupois. 

9.  Find  the  length  of  a  centimeter  in  inches. 

10.  Pind  the  number  of  pints  in  a  dekaliter. 

11.  Eeduce  36^^  to  bushels,  pecks,  quarts,  and  pint& 

12.  Eeduce  40.0973^™  to  miles,  rods,  feet,  and  incheii 

13.  Eeduce  12^^  to  Troy  Weight. 

14.  Eeduce  250^^  to  Avoirdupois  Weight. 

15.  Eeduce  to  pounds  Avoirdupois  and  the  decimal  of  a 
pound  T"  4^«  1658. 

16.  How  many  kilometers  make  a  mile  ? 

17.  How  many  hektars  make  a  square  mile  ? 

18.  Eeduce  80  A.  40  sq.  rd.  to  hektars. 

19.  Eeduce  87  bu.  3  pk.  4  qt.  to  hektoliters. 

20.  Eeduce  68|  yd.  to  meters. 

21.  How  many  liters  are  there  in  6  gal.  of  water  ? 

22.  How  many  meters  are  there  in  25  ft.  ? 

23.  2  gal.  3  qt.  \\  pt.  equal  how  many  liters  ? 

24.  2  mi.  17  ft.  16.2  in.  equal  how  many  kilometers  ? 

25.  3  mi.  2  rd.  10  ft.  equal  how  many  meters  ? 

26.  How  many  kilometers  are  there  in  2  mi.  6  fur.  39  rd. 
5  yd.? 

27.  How  many  hektoliters  are  there  in  57  gal.  3^  pt.  ? 

28.  If  the  distance  between  two  places  is  1^  miles,  what 
is  the  number  of  kilometers  ?  of  centimeters  ? 

29.  Find  the  equivalent  of  the  rod  in  the  hektometers. 


THE   METRIC   SYSTEM.  127 

30.  A  lead  pencil  is  If  ^"^  long ;  32186  of  them  arranged 
in  a  line  would  extend  how  many  miles  ? 

31.  Find  the  weight  in  grams  of  a  quart  of  water. 

32.  Find  the  weight  in  grams  of  a  cubic  yard  of  water. 

33.  Find  the  number  of  kilograms  in  a  cubic  foot  of  water. 

34.  Find  the  weight  in  kilos  of  15  gal.  of  water. 

35.  A  cubic  foot  is  what  part  of  a  cubic  meter  ? 

36.  Find  the  weight  in  kilograms  of  a  block  containing 
12516  cu.  in.,  assuming  the  weight  of  a  cubic  inch  of  the 
material  to  be  2  oz. 

37.  A  train  is  27  min.  in  passing  through  a  tunnel,  the 
length  of  which  is  11220"' ;  find  the  speed  of  the  train  in 
miles  per  hour. 

38.  A  platform  bears  a  weight  of  100  lb.  per  square  foot ; 
what  is  the  weight  in  kilograms  per  square  meter  ? 

39.  Find  in  acres  the  area  of  a  plot  300"  long  and  2°" 

wide. 

40.  How  many  square  rods  are  there  in  a  field  300"  long 
and  yig-  of  a  kilometer  wide  ? 

41.  How  many  cubic  yards  are  there  in  a  cistern,  the 
dimensions  of  which  are  64'^'",  225'"",  and  3.75"  ? 

42.  How  many  hektoliters  of  grain  will  a  bin  hold  whose 
interior  length,  width,  and  depth  are  each  6  ft.  6  in.  ? 

43.  Find  the  value  of  a  pile  of  wood  4.5"  long,  1.4"  wide, 
and  1.8"  high,  at  $3.75  a  cord. 

44.  Find  the  value  of  a  pile  of  wood  15  ft.  long,  4  ft. 
wide,  and  4^  ft.  high,  at  f  1.10  a  ster. 

45.  Albany,  IT.  Y.,  is  in  42°  39'  50",  and  Montreal  in  45° 
31'  north  latitude;  find  the  distance  between  them  in 
kilometers. 


128  ARITHMETIC. 

92.  When  an  equivalent  is  given  in  a  problem,  that 
equivalent,  and  no  other,  should  be  used  in  the  solution. 
In  some  cases  the  equivalent  is  one  not  given  in  the  pre- 
ceding section.  Other  problems  are  given  to  show  that  it 
is  possible  to  make  all  such  reductions  by  using  merely  the 
two  equivalents,  1™=  39.37  in.  and  1^=  15.432  gr. 

EXAMPLES. 

1.  The  distance  from  Boston  to  Albany  is  320^" ;  find 
the  distance  in  miles,  assuming  the  meter  to  equal  3-^  ft. 

2.  If  one  kilometer  equals  five  eighths  of  a  mile,  how 
many  turns  will  a  wheel  make  in  20  mi.,  the  circumference 
of  the  wheel  being  4"  5'°'^  ? 

3.  The  deciliter  is  0.026  of  a  gallon ;  wha,t  will  be  the 
weight  in  grams  of  a  pint  of  water  ? 

4.  One  dekagram  is  0.3527  oz.  Avoirdupois ;  how  many 
pounds  Avoirdupois  are  there  in  a  quintal  ? 

5.  The  kilogram  equals  2  lb.  8  oz.  3  pwt. ;  how  many 
centigrams  equal  one  grain  ? 

6.  What  fractional  part  of  -^  of  the  Avoirdupois  ton  is 
12^K,  the  ounce  being  equal  to  28.35^  ? 

7.  A  tunnel  is  2  mi.  21  ch.  13.2  yd.  long.  Find  its  length 
in  meters  (1  mi. =1.61^™). 

8.  If  a  pound  Avoirdupois  equals  0.4536^^,  how  many 
grains  are  there  in  3f^  ? 

9.  A  dekaliter  is  2.6  gal.  What  will  be  the  weight  in 
grams  of  a  quart  of  distilled  water  at  its  greatest  density  ? 

10.  If  a  quart  is  ||-  of  a  liter,  how  many  quarts  are  there 
in  a  box  2"  long,  17'*"  wide,  and  80*^™  deep  ? 

11.  Given  that  a  meter  equals  3.2809  ft. ;  find  how  many 
square  meters  there  are  in  1000  sq.  yd. 


THE   METRIC   SYSTEM.  129 

12.  Find  the  number  of  cubic   inches   (to  the  nearest 
*nith)  in  the  British  imperial  gallon,  which  contains  10  lb. 

A  water.     (Given  l^^-^zzrSS.S  cu.  ft.  and  1*^k=2.2  lb.) 

13.  The  gram  contains  15.432  gr.  How  many  pounds 
Avoirdupois  make  a  myriagram  ? 

14.  The  meter  equals  39.37  in. ;  compare  the  kilometer 
with  the  mile. 

15.  The  meter  equals  39.37  in. ;  compute  from  this  datum 
the  value  of  4  mi.  in  kilometers. 

16.  The  meter  equals  39.37  in. ;  express  in  the  metric 
system  1  ft.  and  1  sq.  ft. 

17.  The  meter  equals  39.37  in. ;  find  how  many  hektars 
make  an  acre. 

18.  One  centimeter  equals  0.3937  in. ;  find  how  many 
cubic  meters  there  are  in  a  cord  of  wood. 

19.  Find  the  weight  in  grams  of  a  cubic  yard  of  water 
(1™=39.37  in.). 

20.  How  many  liters  are  there  in  10  gal.  3  qt.  1  pt.  3  gi., 
the  gallon  being  231  cu.  in.,  and  the  meter  39.37  in.  ? 

21.  A  cubical  vat  measures  9  ft.  in  each  direction  ;  what 
is  its  capacity  in  liters  ?     (Given  1™=39.37  in.) 

22.  How  many  liters  are  contained  in  a  cubical  box  13  in. 
long,  13  in.  wide,  and  13  in.  deep  on  the  inside  ?  (Given 
l'»= 39.37  in.)  How  many  grams  of  water  will  such  a  box 
hold? 

MISCELLANEOUS  EXAMPLES. 

1.  A  tank  can  be  emptied  in  86  min.  by  a  pipe  flowing 
at  the  rate  of  10.8^  a  minute  ;  find  the  value  of  the  contents 
of  the  tank  when  filled  with  oil  worth  $15.25  a  hektoliter. 


130  '  ARITHMETIC. 

2.  If  a  ream  of  paper  is  11.76°'"  thick,  find  the  thickness 
in  millimeters  of  a  single  sheet. 

3.  A  speculator  bought  18.54*  of  land  for  $2500,  and 
sold  it  for  $4.50  a  square  meter ;  find  the  amount  he  gained. 

4.  The  distance  between  two  places  measured  on  a  map 
is  156'"'".  What  is  the  distance  in  kilometers  if  the  scale 
of  the  map  is  1  to  80000  ? 

5.  The  scale  of  a  map  is  4*^""  to  a  hektometer.  The  dis- 
tance between  two  points,  measured  on  the  map,  is  354.2'"'" ; 
what  is  the  actual  distance  between  the  points  in  kilometers  ? 

6.  The  area  of  a  court  is  50.7**1 "'.  How  many  square 
slabs  of  marble,  each  150"^  *'"'  on  the  surface,  will  pave  it  ? 

7.  A  man  bought  a  piece  of  land  52.5"*  square  for  $120  y 
at  what  price  per  ar  must  it  be  sold  to  gain  $56.40  ? 

8.  A  man  bought  30"  of  cloth  at  $2.50  per  meter;  at 
what  price  per  yard  must  it  be  sold  to  gain  $25  ? 

9.  A  man  buys  454  bu.  of  wheat  for  $3  a  bushel,  and 
sells  the  wheat  at  $8.75  a  hektoliter;  how  much  does  he 
gain? 

10.  A  merchant  buys  2|Hm  of  silk  for  $480,  and  sells  the 
silk  at  $1.95  a  yardj  does  he  gain  or  lose,  and  how  much  ? 

11.  If  a  man  pays  600  francs  for  75^  of  wine,  what  is  the 
price  per  gallon  in  United  States  money  ? 

12.  A  field  is  5^^™  long  and  300"°  wide ;  find  its  area  in 
hektars  and  in  acres. 

13.  Find  how  many  cubic  centimeters  there  are  in  a  deka- 
liter of  water.  Find  also  how  many  pounds  Avoirdupoi'i 
there  are  in  the  same  water. 

14.  The  dimensions  of  a  box  are  3.1'",  1.5™,  and  0.6™; 
what  is  the  contents  in  cubic  yards  ?   also  in  dekasters  ? 


THE  METRIC   SYSTEM.  181 

15.  A  rectangular  vessel  is  7™  long,  4.5'*"'  wide,  and  20'^'" 
deep;  find  its  capacity  in  cubic  meters,  hektoliters,  and 
gallons. 

16.  A  vessel  is  3*™  long,  20^™  wide,  and  100""°  deep ;  how 
many  liters  of  water  will  it  contain  ?  How  many  grams  ? 
How  many  cubic  inches  ?     How  many  pounds  ? 

17.  The  water  contained  in  a  vessel  2*"  long,  30°°  wide, 
and  300™""  deep,  would  weigh  how  many  kilograms  ?  would 
measure  how  many  cubic  inches  ?   how  many  gallons  ? 

18.  A  cistern  is  4™  long,  24*"  wide,  and  80""  deep. 

(i)     How  much  water  will  it  hold  in  cubic  meters  ? 
(ii)    How  much  in  liters  1 

(iii)  Find  approximately  the  amount  in  gallons, 
(iv)  What  would  the  contents  weigh  in  kilograms  » 


132  ARITHMETIC. 


CHAPTER   VII. 

SPECIAL  PROBLEMS. 

Carpeting  Rooms. 

93.  Carpeting  comes  in  rolls,  and  is  sold  by  the  yard  or 
meter.  In  estimating  the  amount  of  carpeting  necessary 
for  a  floor,  we  must  ascertain  the  number  of  strips,  and  then 
multiply  the  length  of  a  strip  by  the  number  of  strips.  In 
making  a  carpet,  the  strips  may  run  either  lengthwise  or 
across  the  room;  the  former  way  is  the  more  usual,  and 
should  be  used  in  solving  problems,  except  when  stated  to 
the  contrary. 

Oil-cloth  and  some  other  materials  for  floors  are  sold  by 
the  square  yard  or  square  meter. 

I.    How  many  yards  of  carpet  |-  yd.  wide  does  it  require 

to  cover  a  floor  22  ft.  long  and  16  ft.  wide  ?     How  much 

will  be  turned  under  ? 

5  y(j  _  ;[7  f^^  It  takes  as  many 

^rf4.      ^if^       1^.,   8       198      o«     4.'  strips  as  1|  ft.  is  con- 

16ft.-ll.ft.  =  16XT«^=-W-=8,8^strips.      ,^.^^^   tj„^^^   i„   16 

22  ft.  X  9  =  198  ft.  =  66  yd.  of  carpet.  ft.,  which  equals  S^V 

g g_8   __  _7  strips.     But  a  strip 

is  never  split ;  hence 

15°*^^^ii^8^8^*''^  ^^i  ^^'  ^^^^^^^  under,  i*  is  necessary  to  buy 
^^  9    strips,    and    then 

turn  under  (or  cut  off)  the  excess.  Since  the  length  of  the  room  is 
22  ft.,  each  strip  is  22  ft.  long ;  for  9  strips  it  takes  9  times  22  ft.,  which 
equals  198  ft.,  or  66  yd. 

Since  9  strips  are  bought  and  only  8^j  strips  are  necessary  to  cover 
the  floor,  the  remainder,  f^  of  a  strip,  is  turned  under  (or  cut  off). 


SPECIAL  PROBLEMS.  133 

Since  the  carpet  is  1|  ft.  wide,  /j  of  a  strip  is  ^5  of  1|  ft.,  which  equals 

I  ft.,  or  10^  in. 

II.  How  many  meters  of  carpet  80"*"  wide  will  it  require 
to  cover  a  floor  5.48™  long  and  4.6™  wide,  if  the  strips  are  to 
run  across  the  room  ? 

.8"')  5.48"*  Since  the  strips  run  across  the  room,  the  width 

6.85  strips.      ^^  *^*®  carpet  runs  in  the  same  direction  as  the 

length  of  the  room,   and  we   must   divide  the 

4.6"*  length  of  the  room  by  the  width  of  the  carpet  to 

*      -  find  the  number  of  strips.    Otherwise  the  work  is 

32.2™  the  same  as  in  the  preceding  example. 

Note.  In  examples  dealing  with  the  common  system  of  weights  and 
measures,  common  fractions  are  used.  In  examples  dealing  with  the 
metric  system,  use  decimal  fractions. 


EXAMPLES. 

1.  How  many  yards  of  carpet  f  of  a  yard  wide  does  it 
require  to  cover  a  floor  17  ft.  long  and  16  ft.  6  in.  wide  ? 

2.  How  many  yards  of  carpet  22  in.  wide  will  it  take  to 
cover  a  floor  16  ft.  by  17^  ft.  ? 

3.  A  person,  purchasing  a  carpet  for  a  room  21  ft.  long 
and  15  ft.  9  in.  wide,  chooses  a  material  which  is  |  of  a  yard 
wide,  and  the  pattern  of  which  is  complete  in  each  yard  of 
length.  How  much  carpet  must  he  buy  in  order  that  the 
pattern  may  be  unbroken  ? 

4.  How  many  yards  Of  carpet  f  of  a  yard  wide  will  it 
require  to  cover  a  floor  18^  ft.  long  and'  14  ft.  wide,  if  the 
strips  are  to  run  across  the  room  ? 

5.  A  room  is  24  ft.  3  in.  long  and  16  ft.  4  in.  wide.  Find 
the  cost  of  covering  it  with  a  carpet  f  of  a  yard  wide  at 
$1.25  per  yard. 


134  ARITHMETIC. 

6.  How  many  yards  of  carpet  J  of  a  yard  wide  will  it 
take  to  cover  a  floor  28  ft.  long  and  17  ft.  9  in.  wide,  if  there 
is  a  waste  of  4  in.  in  each  strip  in  matching  patterns  ?  How 
much  will  be  turned  under  ? 

7.  How  many  yards  of  carpet  1^  yd.  wide  does  it  require 
to  cover  a  floor  22  ft.  8  in.  long  and  13  ft.  6  in.  wide,  if  the 
strips  run  across  the  room  ?  How  much  will  be  turned 
under  ?     Find  the  cost  of  the  carpet  at  69  cts.  a  yard. 

8.  How  many  meters  of  carpet  90'^'"  wide  does  it  require 
for  a  floor  7™  long  and  5.4™  wide  ? 

9.  How  many  meters  of  carpet  70*='"  wide  will  it  take  to 
cover  a  floor  5.56™  long  and  4.7™  wide,  if  the  strips  are  to 
run  across  the  room  ? 

10.  How  many  meters  of  carpet  75*^"  wide  will  it  require 
to  cover  a  floor  6™  long  and  4.8™  wide  ?  How  much  will  be 
turned  under  ? 

11.  How  many  meters  of  carpeting  75*=™  wide  will  be 
needed  for  a  room  5|™  square  ?  How  wide  a  strip  must  be 
turned  under  ? 

12.  A  floor  10™  by  6.5™  is  to  be  covered  with  a  carpet 
90*=™  wide  ;  find  the  cost  at  |1.25  per  meter. 

13.  The  floor  of  a  room  is  5.25™  by  4.75™ ;  the  carpet  is 
75cm  ■y^ri(Je  and  is  f  4.25  a  meter.  Find  the  cost  of  the  carpet 
if  3™  is  wasted  in  matching  the  pattern. 

14.  How  many  meters  of  carpet  9*^™  wide  will  cover  a 
floor  6™  long  and  5™  4*^™  wide  ?  What  would  be  the  cost  of 
the  carpet  at  $2.50  a  centar  ? 

15.  It  takes  54  yd.  2\  ft.  of  carpet  to  cover  a  floor  23|-  ft. 
long  and  15f  ft.  wide ;  find  the  width  of  the  carpet. 

16.  It  takes  37.5™  of  carpet  to  cover  a  floor  6.25™  by  5.1".; 
find  the  width  of  the  carpet. 


SPECIAL  PROBLEMS.  135 

17.  What  will  it  cost  to  floor  a  room  17-J-  ft.  long  and 
16  ft.  wide  at  J^l.lO  per  square  yard  ? 

18.  Find  the  cost  of  covering  with  oil-cloth  a  floor  7.56" 
long  and  5.5™  wide  at  62^  cts.  a  square  meter. 

19.  Eind  the  cost  of  covering  with  linoleum  a  floor  19^  tt. 
square  at  75  cts.  a  square  yard. 


Plastering  Booms. 

94.  In  estimating  the  amount  of  plastering  for  a  room, 
we  must  take  the  area  of  the  ceiling  and  the  four  walls,  and 
from  it  subtract  the  area  of  the  doors,  windows,  etc.  The 
area  of  the  four  walls  is  the  same  as  that  of  a  rectangle 
whose  dimensions  are  the  perimeter  and  height  of  the  room. 

Note.  The  length  of  a  base-board  equals  the  perimeter  of  the  room 
minus  the  width  of  the  doors. 

I.  Mnd  the  cost  of  plastering  a  room  32  ft.  long,  21  ft. 
wide,  and  9^  ft.  high,  if  21  sq.  yd.  be  allowed  for  doors  and 
windows,  at  33  cts.  per  square  yard. 

Twice  the  length  added 
to  twice  the  width  equals 
^'•^^  the  perimeter,  and  this 
multiplied  by  the  height 
equals  the  number  of 
square  feet  in  the  four 
walls.  The  area  of  the 
ceiling  equals  the  prod- 
uct of  the  length  and 
breadth.  Adding  these 
tdK'i  r.        J  two  results  we  have  1679 

1651  sq.  yd.  sq.  ft.,   or   186Mq.  yd. 

Subtracting  from  this  21  sq.  yd.,  the  allowance  for  doors  and  windows, 
we  have  165|  sq.  yd.  If  one  square  yard  costs  33  cts.,  165|  sq.  yd.  cost 
165|  times  33  cts.,  which  equals  $54.63. 


64 

32 

$.33 

42 

21 

165f 

106 

32 

9)165 

^i 

64 

18i 

53 

672  sq.  ft. 

165 

954 

1007 

198 

1007  sq.  ft. 

9)1679  sq.ft. 

33 

1864  sq.  yd. 

$54.63 

21 

136  ARITHMETIC. 


EXAMPLES. 

1.  How  many  square  yards  of  plastering  are  there  in  a 
^oom  18  ft.  8  in.  long,  14  ft.  6  in.  wide,  and  9  ft.  high,  mak- 
ing no  allowance  for  doors  and  windows  ? 

2.  How  many  square  meters  of  plastering  are  there  in  a 
room  6.4™  long,  4.8™  wide,  and  3™  high,  making  no  allowance 
for  doors  and  windows  ? 

3.  Find  the  cost  of  plastering  a  room  28  ft.  8  in.  long, 
18  ft.  wide,  and  10  ft.  high,  if  19  sq.  yd.  be  allowed  for  doors 
and  windows,  at  30  cts.  a  square  yard. 

4.  Find  the  cost  of  plastering  a  room  8.4™  long,  5.2™  wide, 
and  3.5™  high,  if  IJ^^  be  allowed  for  doors  and  windows,  at 
38  cts.  a  square  meter. 

5.  Find  the  cost  of  plastering  the  walls  of  a  room  12  ft. 
11  in.  square  and  9  ft.  3  in.  high,  allowing  for  two  windows 
and  one  door  each  6  ft.  2  in.  by  2  ft.  4  in.,  at  28  cts.  a  square 
yard. 

6.  Find  the  cost  of  plastering  a  room  6.4™  square  and  3.8™ 
high  at  42  cts.  a  square  meter.  There  is  a  base-board  30*^™ 
high,  and  an  allowance  of  13.6"^™  is  made  for  doors  and 
windows. 

7.  Find  the  cost  of  plastering  a  room  17  ft.  4  in.  long, 
15  ft.  4  in.  wide,  and  10  ft.  6  in.  high,  at  35  cts.  a  square 
yard.  Make  allowance  for  a  door  8  ft.  by  3  ft.  6  in.,  three 
windows  each  5  ft.  6  in.  by  3  ft.,  and  a  base-board  1  ft.  4  in. 
liigh. 

8.  Find  the  cost  of  plastering  the  walls  of  a  room  10.5™ 
long,  8.4™  wide,  and  4.2™  high,  at  45  cts.  a  square  meter. 
Make  allowance  for  a  door  2.9™  by  1.2™,  two  windows  each 
2.4™  by  1™,  and  a  base-board  32''™  high. 


SPECIAL  PROBLEMS.  137 


Papering  Rooms. 

95.  Paper  is  generally  sold  by  the  roll.  To  find  the 
number  of  rolls  necessary  for  a  room,  we  must  divide  the 
area  of  the  walls  by  the  area  of  one  roll.  When  the  ceiling 
is  to  be  papered,  its  area  should  be  added  to  the  area  of  the 
walls. 

I.  How  many  rolls  of  paper,  7.6™  long  and  50*""  wide,  will 
be  required  for  a  room  6.4*"  long,  5.2™  wide,  and  3.5"  high, 
deducting  14*1™  for  doors  and  windows  ?  Find  the  cost  of 
papering  the  room  at  90  cts.  a  roll  and  of  putting  on  a 
border  at  12  cts.  a  meter. 

$.90  f  .12  The  area  of  the 

18  23.2     walls  is  found  to  be 


12.8 

7.5 

10.4 

.50 

23.2 

3.750 

3.5 

1160 
696 

3.75)67.20(17  + 
375 

81.20 
14. 

2970 
2625 

67.2'«i™            ^45 

18  rolls. 

16.20     f  2.784     67.2«i'n,and  the  area 
2.78  of  one  roll  of  paper 

$18  98  ^^  3.75*«™.  It  will  require  as 
many  rolls  as  3.75*i  ™  is  con- 
tained times  in  67.2"^™,  whicli 
equals  17  and  a  fraction.  But 
a  fraction  of  a  roll  is  never 
sold ;  hence  it  is  necessary  to 
buy  18  rolls.  If  1  roll  costs 
90  cts.,  18  rolls  cost  18  times 
90  cts.,  which  equals  $16.20.  The  border  runs  around  the  room,  and  its 
length  equals  the  perimeter  of  tlie  room.  If  1  meter  costs  12  cts.,  23.2"^ 
cost  23.2  times  12  cts.,  which  equals  $2.78.  The  entire  cost  is  the  sum 
of  116.20  and  $2.78,  which  equals  $18.98. 


EXAMPLES. 

1.  How  many  rolls  of  paper,  7^  yd.  long  and  18  in.  wide, 
will  be  required  for  a  room  26  ft.  long,  20  ft.  6  in.  wide, 
and  9  ft.  4  in.  high,  deducting  17  sq.  yd.  for  doors  and 
windows  ? 


138  ARITHMETIC. 

2.  How  many  rolls  of  paper,  8*"  long  and  45^""  wide,  Avill 
be  required  for  a  room  T.S""  long,  5.4'"  wide,  and  3™  high, 
deducting  21"^  ™  for  doors  and  windows  ? 

3.  Find  the  cost  of  papering  a  room  23  ft.  8  in.  long, 
20  ft.  6  in.  wide,  and  10  ft.  high,  with  paper,  each  roll  of 
which  is  8  yd.  long  and  18  in.  wide,  at  62-^  cts.  a  roll,  allow- 
ing for  a  door  7  ft.  6  in.  by  3  ft.  3  in.,  and  two  windows 
each  5  ft.  8  in.  by  3  ft.  3  in. 

4.  Find  the  cost  of  papering  a  room  7.5™  square  and 
3.5™  high,  with  paper,  each  roll  of  which  is  8"'  long  and  50*=" 
wide,  at  $1.10  a  roll,  and  of  putting  on  a  border  at  15  cts.  a 
meter.  Make  allowance  for  two  doors  each  2.8™  by  1.2™, 
two  windows  each  2.2™  by  1™,  and  a  base-board  30''™  high. 

5.  Find  the  cost  of  papering  a  room  19|-  ft.  long,  16J  ft. 
wide,  and  10|-  ft.  high,  with  paper,  each  roll  of  which  is 
7^  yd.  long  and  20  in.  wide,  at  80  cts.  a  roll,  and  of  putting 
on  a  border  at  12|-  cts.  a  yard.  Make  allowance  for  a  door 
8  ft.  by  S^  ft.,  three  windows  each  5|-  ft.  by  3  ft.,  and  a  base- 
board 1^  ft.  high. 

6.  Find  the  cost  of  papering  the  walls  and  ceiling  of  a 
room  11.2™  long,  9.6™  wide,  and  5™  high,  with  paper,  each 
roll  of  which  is  7.5™  long  and  50*=™  wide,  at  $1.25  a  roll. 
Make  allowance  for  two  doors  each  3.5™  by  1.5™,  three  win- 
dows each  2.5™  by  1.4™,  and  a  base-board  40"''"  high. 

7.  How  many  yards  of  paper  1^  yd.  wide  will  be  needed 
to  paper  the  walls  of  a  room  10  ft.  high,  18  ft.  long,  and 
12  ft.  wide  ? 

8.  Find  the  cost  of  papering  a  room  11  yd.  2  ft.  4  in. 
long,  6  yd.  2  ft.  wide,  and  5  yd.  2  ft.  6  in.  high,  with  paper 
1  yd.  4  in.  wide,  at  6  cts.  a  yard. 

9.  Find  the  cost  of  papering  a  room  6.78™  long,  4.1™ 
vade,  and  3.64™  high,  with  paper  96*=™  wide,  at  8  cts.  a 
meter. 


SPECIAL   PROBLEMS.  139 


Board  Measure. 

96.  The  standard  thickness  for  boards  is  one  inch.  A 
board  one  inch  or  less  in  thickness  contains  as  many  board 
feet  as  there  are  square  feet  in  its  surface.  For  boards  more 
than  one  inch  thick,  and  for  other  kinds  of  lumber,  we  must 
multiply  the  number  of  square  feet  in  the  surface  by  the 
number  of  inches  in  thickness  in  order  to  find  the  number 
of  board  feet.  For  example,  a  board  8  ft.  long,  1  ft.  wide, 
and  1  in.  or  less  in  thickness,  contains  8  board  feet ;  a  plank 

8  ft.  long,  1  ft.  wide,  and  3  in.  thick,  contains  24  board  feet. 
When  the  metric  system  is  used,  the  standard  thickness 

is  25""".  A  board  25™'"  or  less  in  thickness  contains  as  many 
board  meters  as  there  are  square  meters  in  its  surface.  For 
boards  more  than  25"""  thick,  and  for  other  kinds  of  lumber, 
we  must  multiply  the  number  of  square  meters  in  the  sur- 
face by  the  number  of  times  25""°  is  contained  in  the  thick- 
ness. For  example,  a  board  4™  long,  50*=™  wide,  and  25"*™  or 
less  in  thickness,  contains  2  board  meters  j  a  plank  4""  long, 
50*=™  wide,  and  10*="  thick,  contains  8  board  meters. 

If  a  board  is  tapering,  we  must  take  the  average  width, 
which  is  one  half  the  sum  of  its  two  end  widths. 

In  buying  and  selling  boards,  it  is  customary  to  quote 
them  by  the  hundred  or  thousand,  meaning  a  hundred  or 
thousand  board  feet,  or  a  hundred  or  thousand  board  meters. 

EXAMPLES. 

1.  How  many  board  feet  are  there  in  a  board  16  ft.  long, 

9  in.  wide,  and  -J  in.  thick  ? 

2.  How  many  board  feet  are  there  in  a  board  17  ft.  6  in. 
long,  1  ft.  3  in.  wide,  and  1  in.  thick  ? 

3.  HoAv  many  feet,  board  measure,  are  there  in  a  plank 
12  ft.  4  in.  long.  2  ft.  3  in.  wide,  and  4  in.  thick  ? 


140  ARITHMETIC. 

4.  How  many  feet,  board  measure,  are  there  in  a  plank 
16  ft.  4  in.  long,  1  ft.  7  in.  wide,  and  4|-  in.  thick  ? 

5.  How  many  feet,  board,  measure,  are  there  in  a  plank 
12  ft.  4  in.  long,  2  ft.  5  in.  wide  at  one  end,  2  ft.  1  in.  wide  at 
the  other,  and  4  in.  thick  ? 

6.  How  many  feet  of  board  are  there  in  a  plank  17  ft. 
long,  22  in.  wide  at  one  end,  13  in.  wide  at  the  other,  and 
3  in.  thick  ? 

7.  Find  the  number  of  board  feet  in  a  stick  of  timber 
18  ft.  long  and  8  in.  square. 

8.  Find  the  cost  of  72  boards,  each  11  ft.  long,  16  in. 
wide,  and  f  in."  thick,  at  $16.50  per  M. 

9.  Find  the  cost  of  14  joists,  each  4  in.  by  3  in.,  and 
10  ft.  long,  at  f  13.75  per  M. 

10.  Find  the  cost  of  the  flooring  for  two  rooms,  each 
24  ft.  by  201  ft.,  with  boards  IJ  in.  thick,  at  f  27  per  M. 

11.  How  many  board  meters  are  there  in  a  board  7"  long, 
18"»  wide,  and  20°^™  thick  ? 

12.  How  many  board  meters  are  there  in  a  board  7.5"* 
long,  20*""  wide,  and  30°*™  thick  ? 

13.  How  many  meters,  board  measure,  are  there  in  a 
plank  5.8"*  long,  30^™  wide,  and  75"*'°  thick  ? 

14.  How  many  meters,  board  measure,  are  there  in  a 
plank  6™  long,  36*^"*  wide  at  one  end,  30*=™  wide  at  the  other, 
and  11*=™  thick  ? 

15.  Find  the  number  of  board  meters  in  a  stick  of  timber 
9™  long  and  30*=™  square. 

16.  Find  the  cost  of  50  boards,  each  4™  long.  36"°  wide, 
and  25™"*  thickf  at  $18  per  C. 


SPECIAL   PROBLEMS.  141 

17.  Find  the  cost  of  24  planks,  each  4.5"  long,  42^  wide, 
and  8«'»  thick,  at  122.50  jjer  C. 

18.  Find  the  cost  of  a  stick  of  timber  10.6"  long,  30'"' 
wide  at  one  end,  25*="  wide  at  the  other,  and  20*^"  thick,  at 
^18.50  per  C. 

Work  Problems. 

97.  Problems  concerning  work  should  be  solved  by  con- 
sidering the  fractional  part  of  the  work  that  can  be  done  in 
a  unit  of  time.  For  example,  if  a  man  can  do  a  piece  of 
work  in  8  days,  he  can  do  -J  of  the  work  in  one  day ;  if  a 

cistern  can  be  filled  by  a  pipe  in  2^  hours,  — ,  or  -j^,  can  be 
filled  in  one  hour.  * 

I.  A  can  do  a  piece  of  work  in  8  days,  working  10  hours 
a  day,  and  B  can  do  it  in  6  days,  working  12  hours  a  day  ; 
in  how  many  days  of  9  hours  each  can  they  together  do  it  ? 

j^        J    _  g^j  Q  _   ^Q  Since  A  can  do  the  work 

TU~^^  —  TTG'  —  rfU-  in  80  hr.,  he  can  do  ^^  of  it 

1  "^  T  A^  =  1  X  ^^  =  37fJ  hr.  in  1  hr. ;  since  B  can  do  it  in 

37i^-F-9  =  ^H-9  =  ff  =  4^da.      72  hr.,  he  can  do  ^V  of  it  in 

1  hr. ;  hence  both  together 
can  do  jV+  tV*  ^^  jts*  ^^  1  ^^-  Since  they  can  do  J^j^  in  1  hr.,  it  will 
take  as  many  hours  to  do  the  whole  as  j^^^  is  contained  times  in  1, 
which  equals  37||  hr.  Since  they  work  9  hr.  a  day,  it  will  take  as 
many  days  as  9  is  coiltained  times  in  37}|,  which  equals  4j\  da. 

II.  A  cistern  can  be  filled  by  a  pipe  in  4^  hours,  and  can 
be  emptied  by  another  pipe  in  6|-  hours ;  if  both  pipes  be 
opened,  in  what  time  will  the  cistern  be  filled  ? 

By  the  first  pipe  /^  of 
the  cistern  will  be  filled  in 

1100111"^  ^  ^^''  ^"^  ^^  *^^  second 

J-  -^  Tlnr  =  1  X  V-  =  1%  ^i"-  pipe  -^  will  be  emptied  in 

the  ^ajne  time;  hence,  when  both  pipes  are  open-  /y  — 2%»  ^^  jwu*  ^1 


116 

3      24-15       9 

^■     6|     25 " 

20         100         100 

142  ARITHMETIC. 

be  filled  in  1  hr.     It  will  take  as  many  hours  to  fill  the  cistern  as  j^  ig 
contained  times  in  1,  which  equals  11^  hr. 


EXAMPLES. 

1.  A  can  mow  a  field  in  4|-  days,  and  B  can  mow  it  in 
6  days ;  how  long  will  it  take  them  both  to  mow  it  ? 

2.  A  can  build  a  wall  in  18f  days,  and  B  in  31:|^  days ; 
how  long  will  it  take  them  both  together  to  build  it  ? 

3.  A  cistern  can  be  filled  by  a  pipe  in  3^  hours,  and  by 
another  in  2^  hours ;  how  long  will  it  take  both  together  to 
fill  it  ? 

4.  A  can  do  a  certain  piece  of  work  in  6  days,  B  in  8 
days,  and  C  in  9  days.  How  long  will  it  take  them  to  do 
it  together  ? 

5.  A  can  dig  a  ditch  in  5  days,  B  in  7  days,  and  C  in  9 
days ;  how  long  will  it  take  them  all  together  to  dig  it  ? 

6.  A  can  do  a  piece  of  work  in  12  days,  B  in  15  days, 
and  C  in  20  days ;  what  fractional  part  of  the  work  can 
they  together  do  in  3  days  ? 

7.  A  cistern  has  two  pipes,  one  of  which  can  fill  it  in  3 
hours,  and  the  other  in  4  hours  ;  a  third  pipe  can  empty  it 
in  2  hours.  If  all  three  are  opened  when  the  cistern  is 
empty,  in  what  time  will  it  be  filled  ? 

8.  A  cistern  can  be  filled  by  a  pipe  in  75  min.,  and  can 
be  emptied  by  another  pipe  in  30  min. ;  if  the  cistern  is 
full,  and  both  pipes  are  open,  in  what  time  will  the  cistern 
be  emptied  ? 

9.  Pipes  A  and  B  can  fill  a  cistern  in  3  min.  and  5  min. 
respectively,  and  C  can  empty  it  in  7^  min.  In  what  timt 
will  the  cistern  be  filled  when  A^  B,  and  C  are  all  open  ? 


SPECIAL  PROBLEMS.  143 

10.  A  and  B  can  do  a  piece  of  work  in  12  days.  A,  work- 
ing alone,  can  do  the  same  work  in  20  days.  How  long 
would  it  take  B  to  do  it  ? 

11.  A  can  do  a  piece  of  work  in  10  days ;  A  and  B  can 
do  the  same  work  together  in  7  days ;  in  how  many  days 
can  B,  working  alone,  do  the  work  ? 

12.  A  can  do  a  piece  of  work  in  10  days,  A  and  C  can  do 
it  in  7  days,  and  A  and  B  can  do  it  in  6  days ;  in  how  many 
days  can  B  and  C  together  do  it  ? 

13.  A  can  do  ^  of  a  piece  of  work  in  4  days,  B  ^  in  5 
days,  C  ^  in  3  days,  and  D  ^  in  1^  days ;  how  long  will  it 
take  them  all  to  do  it  ? 

14.  Three  men  can  do  a  piece  of  work  in  12  hours ;  A 
and  B  can  do  it  in  16  hours,  and  A  and  C  in  18  hours. 
What  part  can  B  and  C  do  in  9^  hours  ? 

15.  A  can  do  a  certain  piece  of  work  in  10  days,  working 
8  hours  a  day.  B  can  do  the  same  work  in  9  days,  working 
12  hours  a  day.  They  decide  to  work  together,  and  to  finish 
the  work  in  6  days.  How  many  hours  a  day  must  they 
work? 

16.  A  does  ^  of  a  piece  of  work  in  6  days,  when  B 
comes  along  and  helps  him,  and  they  finish  it  in  5  days ; 
how  long  would  it  take  B  alone  to  do  the  work  ? 

17.  A  can  do  as  much  work  in  4  hours  as  B  in  6,  and  B 
in  3^  as  C  in  5.  A  does  half  a  certain  piece  of  work  in  12 
hours ;  in  what  time  can  it  be  finished  by  B  and  C,  working 
separately  equal  times  ? 

18.  A  and  B  can  do  a  piece  of  work  in  6  days,  A  and  C 
in  7  days,  and  B  and  C  in  8  days;  in  what  time  can  all 
three  do  it,  working  together^  and  in  what  time  can  each 
one  do  it  alone  ? 


144  ARITHMETIC. 

Clock  Problems. 

98.  In  twelve  hours  the  minute  hand  of  a  clock  passes 
over  the  face  twelve  times,  and  if  the  hour  hand  were  sta- 
tionary, the  two  hands  would  be  together  twelve  times. 
But  in  this  interval  of  twelve  hours,  the  hour  hand,  instead 
of  remaining  stationary,  passes  over  the  face  once ;  hence 
the  two  hands  are  together  once  less  than  twelve  times,  or 
eleven  times.  Since  the  two  hands  progress  with  a  regular 
movement,  there  always  is  the  same  interval  between  two 
successive  times  when  the  hands  are  together,  and  this  in- 
terval is  ^  of  12  hours,  which  equals  1  hr.  5^  min.  It  is 
also  true  of  any  other  position  of  the  hands,  that  there  is  an 
interval  of  1  hr.  5^  min.  between  two  successive  times  when 
the  hands  have  the  same  relative  position. 

I.  At  what  time  betw-een  7  and  8  o'clock  are  the  hands 
of  a  clock  together  ? 

1  hr.     5^  min.  The  hands  are  together  at  12  o'clock,  and  there 

7  are  7  intervals  from  12  o'clock  to  the  required  time. 


7  hr   38  2   min        Since  each  interval  is  1  hr.  5y\  min.,  7  intervals 
equal  7  times  1  hr.  5y\  min.,  which  equals  7  hr. 
Ans.   38  j^  min.    33^2_  min.     Hence  the  required  time  is  38i\  min. 
past  7  o'clock,  past  7  o'clock. 

II.   At  what  time  between  5  and  6  o'clock  are  the  hands 
of  a  clock  at  right  angles  ? 

Ihr.     5,5.  min.  1  hr.     S^-^min.  The  hands  are  at  right 

2  Q  angles  at  3  o'clock  and  at 

9  o'clock.    There  are  2  in- 


o       '        ^  -^  '  Q       '        ^  ^  tervals  from  3  o'clock  to  the 

i:r-, TTTTT r—  tt; tttz : —      required    time;     hence   we 

5  hr.  lOi^  mm.         5  hr.  43,^,  mm.     ^^^  ^  j^^    ^^,,  ^.^  ^^  3 

Ans.  lOfJ  min.  past  5  o'clock,         o'clock   to  obtain  one  an- 

or  43iV  min.  past  5  o'clock.         ^^^^-  ^^''^  ^'^  ^  i"*^^^^^« 

from  9  o'clock  to  the  re- 
quired time ;  hence  we  add  8  hr.  43j'^y  min.  to  9  o'clock  to  obtain  the  other, 
answer,  writing:  5  hr.  instead  of  V  hr..  because  17  ig  j5  pjore  than  12- 


SPECIAL  PROBLEMS.  145 

III.   Find  when  first  after  1  o'clock  the  hands  of  a  clock 

make  an  angle  of  60°  with  each  other. 

^  g       .  The  hands  of  a  clock  make 

1  hr.    Oyy  mm.  ^^  ^^^^^  ^^  g^o  ^i^^  each  other 

at  2  o'clock  and  at   10  o'clock. 


3  hr.  16y\  min.  When  first  after  1  o'clock  they 
10  make  an  angle  of  60°  with  each 
Z     7^~4       '-                                     other,  they  have  the  same  rela- 

1  hr.  16yy  mm.  ^j^^  position  as  at  10  o'clock. 

A71S.   16y\  min.  past  1  o'clock.      There   are  3  intervals   from   10 

o'clock  to  the  required  time; 
hence  we  add  .3  hr.  16^^  min.  to  10  o'clock  to  obtain  the  answer. 


EXAMPLES. 

1.  At  what  time  between  4  and  5  o'clock  are  the  hands 
of  a  clock  together  ? 

2.  At  what  time  between  6  and  7  o'clock  are  the  hands 
of  a  clock  together  ? 

3.  At  what  time  between  8  and  9  o'clock  are  the  hands 
of  a  clock  together  ? 

4.  At  what  time  between  12  and  1  o'clock  are  the  hands 
of  a  clock  opposite  each  other  ? 

5.  At  what  time  between  3  and  4  o'clock  are  the  hands 
of  a  clock  opposite  each  other  ? 

6.  At  what  time  between  9  and  10  o'clock  are  the  hands 
of  a  clock  opposite  each  other  ? 

7.  At  what  time  between  2  and  3  o'clock  are  the  hands 
of  a  clock  at  right  angles  ? 

8.  At  what  time  between  4  and  5  o'clock  are  the  hands 
of  a  clock  at  right  angles  ?  ^ 

9.  At  what  time  between  9  and  10  o'clock  are  the  hands 
of  a  clock  at  risrht  an^le^  : 


146  ARITHMETIC. 

10.  At  what  time  between  11  and  12  o'clock  are  the  hands 
of  a  clock  at  right  angles  ? 

11.  Find  when  first  after  11  o'clock  the  hands  of  a  clock 
make  an  angle  of  30°  with  each  other. 

12.  Find  when  first  after  2  o'clock  the  hands  of  a  clock 
make  an  angle  of  60°  with  each  other. 

13.  Find  when  first  after  6  o'clock  the  hands  of  a  clock 
make  an  angle  of  120°  with  each  other. 

14.  Find  when  first  after  4  o'clock  the  hands  of  a  clock 
make  an  angle  of  150°  with  each  other. 

Comparison  of  Thermometers. 

99.  There  are  two  important  points  to  be  determined  in 
the  graduation  of  a  thermometer,  —  the  freezing  point  and 
the  boiling  point  of  water. 

In  the  Fahrenheit  scale  the  freezing  point  is  marked  32°, 
and  the  boiling  point  212° ;  the  intervening  space  is  divided 
into  180  equal  parts  called  degrees. 

In  the  Centigrade  scale  the  freezing  point  is  marked  0°, 
and  the  boiling  point  100° ;  the  intervening  space  is  divided 
into  100  degrees. 

In  the  E^aumur  scale  the  freezing  point  is  marked  0°, 
and  the  boiling  point  80°  ;  the  intervening  space  is  divided 
into  80  degrees. 

In  expressing  temperatures  it  is  customary  to  indicate 
the  scale  referred  to  by  the  initial  letters  F.,  C,  and  R. 

Temperatures  below  0°  are  indicated  by  the  minus  sign. 
For  example,  -15°  C.  indicates  15°  below  0° ;  -10°  F.  indi- 
cates 10°  below  0°  or  42°  below  the  freezing  point. 

Since  180  Fahrenheit  degrees  =  100  Centigrade  degrees 
=  80  Reaumur  degrees,  9  Fahrenheit  degrees  =  5  Centi- 
grade degrees  =  4  Reaumur  degreee 


SPECIAL  PROBLEMS.  147 

I.  Express  95°  T.  in  Centigrade  scale. 

95  7  Subtracting  32  from  95,  we  find  that  95° 

32     2  of  ^^  _  350  Q  p   jg  53  degrees  above  the   freezing   point. 

63     '^  Since  9  Fahrenheit  degrees  =  5  Centigrade 

degrees,  there  are  |  as  many  Centigrade  degrees  as  Fahrenheit  degrees; 
hence  95°  F.  is  35°  above  the  freezing  point  in  the  Centigrade  scale,  or 
36°  C. 

II.  Express  45°  C.  in  the  Fahrenheit  scale. 

Since  9  Fahrenheit  degrees  =  5  Centigrade 

9    -J«_Q-i  degrees,  there  are  |  as  many  Fahrenlieit  de- 

^  *  grees  as  Centigrade  degrees ;   hence  45°  C.  is 

81  -4-  32  =  113°  F  ^^°  above  the  freezing  point  in  the  Fahrenheit 

*  scale,  or  113°  F. 

EXAMPLES. 

1.  Express  86°  F.  in  the  Centigrade  scale. 

2.  Express  68°  F.  in  the  Centigrade  scale. 

3.  Express  23°  F.  in  the  Centigrade  scale. 

4.  Express  —4°  F.  in  the  Centigrade  scale. 

5.  Express  60°  C.  in  the  Fahrenheit  scale. 

6.  Express  15°  C.  in  the  Fahrenheit  scale. 

7.  Express  —10°  C.  in  the  Fahrenheit  scale. 

8.  Express  —30°  C.  in  the  Fahrenheit  scale. 

9.  Express  50°  F.  in  the  Reaumur  scale. 

10.  Express  14°  F.  in  the  E^amnur  scale. 

11.  Express  24°  R.  in  the  Fahrenheit  scale. 

12.  Express  —16°  R.  in  the  Fahrenheit  scale. 

13.  Express  60°  C.  in  the  Reaumur  scale. 

14.  Express  —25°  C.  in  the  Reaumur  scale. 

15.  Express  36°  R.  in  the  Centigrade  scale, 

16.  Express  — 16"  R.  in  the  Centigrade  scale. 


148  ARITHMETIC. 


Specific  Gravity. 

100.  The  specific  gravity  (sp.  gr.)  of  any  substance  is  its 
weight  compared  with  the  weight  of  an  equal  bulk  of  water. 
Since  water  is  the  standard,  its  specific  gravity  is  1.  The 
specific  gravity  of  any  other  substance  denotes  the  number 
of  times  it  is  heavier  than  water.  For  example,  if  a  bar  of 
silver  has  a  specific  gravity  of  10.5,  it  is  10.5  times  as  heavy 
as  an  equal  bulk  of  water. 

In  the  metric  system  of  weights  and  measures,  the  weight 
of  any  bulk  of  water  can  readily  be  found  by  remembering 
that  1  cubic  meter  of  water  weighs  1  metric  ton,  1  liter 
weighs  1  kilogram,  and  1  cubic  centimeter  weighs  1  gram. 
In  the  common  system  of  weights  and  measures,  1  cubic 
foot  of  water  weighs  1000  ounces  Avoirdupois. 

When  the  bulk  and  specific  gravity  of  a  substance  are 
known,  the  weight  of  the  substance  can  be  found  by  multi- 
plying the  weight  of  an  equal  bulk  of  water  by  the  specific 
gravity. 

When  the  bulk  and  weight  of  a  substance  are  known,  the 
specific  gravity  of  the  substance  can  be  found  by  dividing 
the  weight  of  the  substance  by  the  weight  of  an  equal  bulk 
of  water. 

When  the  weight  and  specific  gravity  of  a  substance  are 
known,  the  weight  divided  by  the  specific  gravity  equals 
the  weight  of  an  equal  bulk  of  water,  and  from  this  the 
bulk  of  the  substance  can  readily  be  found. 

All  bodies  weigh  less  in  water  than  in  air.  It  can  be 
proved  by  experiment  that  the  difference  between  the  weight 
of  a  body  in  air  and  its  weight  in  water  equals  the  weight 
of  the  water  displaced.  Hence,  if  the  weight  of  a  body  in 
air  be  divided  by  the  difference  between  its  weight  in  air 
and  its  weight  in  water,  the  Yesult  is  the  specific  gravity. 


SPECIAL  PROBLEMS. 


149 


J.   Find  the  weight  of  a  bar  of  copper  (sp.  gr.  8.79)  2  ft. 
long  and  3  in.  square. 


?xlxi  =  |cu.ft. 

2 

,  125 

i  of  xm=   125  oz. 

1125 
875 
1000 


The  dimensions  expressed  in  feet  are  2  ft., 
\  ft.,  and  \  ft.,  and  the  product  of  these  di- 
mensions gives  ^  cu.  ft.  as  the  cubic  contents 
The  weight  of  an  equal  bulk  of  water  is  |  of 
1000  oz,,  or  125  oz.  Multiplying  this  result 
by  8.79,  we  find  the  weight  of  the  copper  to 
be  1098.75  oz.,  which  equals  68.672  lb. 


16)1098.75  oz. 
68.672  lb. 


II.   If  650'="*='"  of  ether  weigh  468«,  what  is  its  specific 
gravity  ? 


650)468.00(0.72 
4550 
1300 
1300 


650cucm  of  water  weigh  650  k;  hence  the  spe- 
cific gravity  of  ether  is  as  much  as  650  is  con- 
tained in  468,  which  equals  0.72. 


III.   Find  the  bulk  of  a  piece  of  coal  (sp.  gr.  1.8)  which 

weighs  56.88  H 

1.8)56.88(31.6 
54 
28 
18 
108 
108 


Since  the  specific  gravity  of  coal  is  1.8,  the 
weight  of  an  equal  bulk  of  water  is  as  much  as 
1.8  is  contained  in  56.88^8,  which  equals  31.6  Kg, 
or  31600  K.  The  bulk  of  31600  k  of  water  is 
31600  c«  cm,  which  is  also  the  bulk  of  the  coal. 


^TlS.  31600  «="«=". 

EXAMPLES. 

1.  Find  the  weight  of  a  cubic  foot  of  ice  (sp.  gr.  0.92). 

2.  Find  the  weight  of  a  gallon  of  milk  (sp.  gr.  1.03). 

3.  Find  the  weight  in  grains  of  a  cubic  inch  of  iron 
(sp.  gr.  7.21). 


150  ARITHMETIC. 

4.  Find  the  weight  of  a  bar  of  platinum  10  in.  long, 
4  in.  wide,  and  1^  in.  thick,  if  its  specific  gravity  is  22.07. 

5.  A  tank  is  6  ft.  long,  4  ft.  wide,  and  3  ft.  deep.     How 
many  pounds  of  sulphuric  acid  (sp.  gr.  1.84)  will  it  contain? 

6.  Find  the  specific  gravity  of  a  stone,  a  cubic  foot  of 
which  weighs  185  lb. 

7.  Find  the  specific  gravity  of  a  liquid  weighing  10  lb. 
per  gallon. 

8.  A  bar  of  gold  3  in.  long,  1^  in.  wide,  and  i  in.  thick 
weighs  25  oz.  Avoirdupois ;  find  its  specific  gravity. 

9.  A  piece  of  iron  weighs  12  lb.  in  air  and  10^  lb.  in 
water ;  find  its  specific  gravity. 

10.  Find  the  number  of  cubic  inches  in  a  pound  of  alu- 
minium (sp.  gr.  2.64). 

11.  Find  the  number  of  bushels  in  a  ton  of  salt  (sp.  gr. 
2.15). 

12.  A  piece  of  glass  weighs  4320  gr.  in  air  and  3195  gr. 
in  water ;  what  is  its  specific  gravity  ?  its  volume  ? 

13.  Find  the  weight  of  58^  of  sand  (sp.  gr.  1.65). 

14.  Find  the  weight  of  6.32  ^^  of  olive  oil  (sp.  gr.  0.915). 

15.  A  plank  is  5°»  long,  3^  wide,  and  3*=°^  thick;  find  its 
weight  in  grams,  if  the  specific  gravity  of  the  wood  is  0.8. 

16.  A  tank  is  1.5™  wide,  3.2™  long,  and  80'""  deep.  How 
many  kilograms  of  alcohol  (sp.  gr.  0.8)  will  be  required  to 
fill  it  one  third  full  ? 

17.  What  is  the  weight  in  metric  tons  of  a  block  of  stone 
(sp.  gr.  2.5)  measuring  12.37"  by  7.14™  by  83*=™? 

18.  Find  the  specific  gravity  of  an  acid  weighing  1.58^* 
per  liter. 


SPECIAL  PROBLEMS.  151 

19.  If  2V  of  alcohol  weigh  22.14^,  what  is  its  specific 
gravity  ? 

20.  A  brick  20"=™  long,  11  "^^  wide,  and  5.5  <'°'  thick  weighs 
2.904  ^« ;  find  its  specific  gravity. 

21.  A  plate  of  iron  137  «="  long,  643""  wide,  and  43"" 
thick  weighs  277.54^^.     What  is  its  specific  gravity? 

22.  A  stone  weighs  8.42  ^^^  in  air  and  6.32  ^^  in  water ;  find 
its  specific  gravity. 

23.  A  body  weighs  460 «  in  air  and  401.16 «^  in  water;  what 

is  its  specific  gravity  ? 

24.  Find  the  number  of  cubic  centimeters  in  a  piece  of 
brass  (sp.  gr.  8.38)  weighing  86.7338. 

25.  If  alcohol  (sp.  gr.  0.81)  costs  $1.45  a  kilogram,  what 
is  the  price  of  a  liter  ? 

26.  If  salt  (sp.  gr.  2.15)  costs  $7.50  a  metric  ton,  what 
is  the  price  of  a  hektoliter  ? 

27.  If  cork  (sp.  gr.  0.24)  is  worth  2\  cts.  a  cubic  deci- 
meter, find  the  value  of  10^«. 

28.  If  marble  (sp.  gr.  2.83)  is  worth  $28.50  a  cubic  meter, 
find  the  value  of  a  block  weighing  764  ^«. 

Longitude  and  Time. 

101.  The  longitude  of  a  place  is  the  arc  or  portion  of  the 
equator  between  a  standard  meridian  and  the  meridian  of 
the  given  place.  A  place  is  in  east  or  west  longitude,  ac- 
v3ording  as  it  is  east  or  west  of  the  standard  meridian,  and 
the  longitude  is  reckoned  in  degrees,  minutes,  and  seconds 
up  to  180°,  or  half  way  round  the  earth.  For  example,  long. 
32°  25'  W.  indicates  a  place  situated  on  the  meridian  which 
is  32°  25'  west  of  the  standard  meridian.     The  meridian  of 


152  ARITHMETIC. 

Greenwich,  England,  is  usually  taken  as  the  standard  by- 
English-speaking  people. 

When  two  places  are  on  the  same  side  of  the  standard 
meridian,  the  difference  of  longitude  is  found  by  subtract- 
ing their  longitudes ;  when  two  places  are  on  opposite  sides 
of  the  standard  meridian,  the  difference  of  longitude  is 
found  by  adding  their  longitudes.  If,  however,  in  the  lat- 
ter case,  the  sum  exceeds  180°,  it  must  be  subtracted  from 
360°  to  obtain  the  correct  difference  of  longitude. 

The  earth  revolves  on  its  axis  once  in  24  hours,  thus 
making  360°  of  longitude  pass  under  the  sun  in  that  time. 
In  1  hr.  J^  of  360°,  or  15°,  pass  under  the  sun ;  in  1  min. 
Jg-  of  15°,  or  15';  in  1  sec.  -^  of  15',  or  15".  Hence  a  dif- 
ference of  15°  of  longitude  causes  a  difference  of  1  hr.  of 
time;  a  difference  of  15'  of  longitude  causes  a  difference 
of  1  min.  of  time ;  a  difference  of  16"  of  longitude  causes 
a  difference  of  1  sec.  of  time. 

I.  The  difference  of  time  between  two  places  is  2  hr.  15 
min.  27  sec. ;  what  is  the  difference  of  longitude  ? 

2  hr.  15  min.  27  sec.  Since  1  hr.  of  time  corresponds  to  15° 

15  of  longitude,  1  min.  of  time  to  15'  of 

330     gY'  45''  longitude,  and  1  sec.  of  time  to  15"  of 

longitude,  15  times  the  number  of  hours, 
minutes,  and  seconds  equals  the  number  of  degrees,  minutes,  and 
seconds. 

II.  The  difference  of  longitude  between  two  places  is 
48°  24'  36" ;  what  is  the  difference  of  time  ? 

15)48°      24'  36"  Since  15°  of  longitude  corresponds 

3hr.l3min.38|sec.     *«  ^  ^^-  «^  ^^^^'  ^^'  «^  longitude  to 
1  min.  of  time,  and  15"  ot   xongitude 
to  1  sec.  of  time,  -^-g  of  the  number  of  degrees,  minutes,  and  seconds 
equals  the  number  of  hours,  minutes,  and  seconds. 


SPECIAL  PROBLEMS.  168 


EXAMPLES. 

1.  The  difference  of  time  between  two  places  is  6  hr. 
42  min.  22  sec. ;  what  is  the  difference  of  longitude  ? 

2.  The  difference  of  time  between  St.  Petersburg  and 
St.  Paul  is  8  hr.  13  min.  36  sec. ;  what  is  the  difference  of 
longitude  ? 

3.  The  difference  of  time  between  Boston  and  St.  Louis 
is  1  hr.  16  min.  47  sec. ;  what  is  the  difference  of  longitude  ? 

4.  The  time  in  Montreal  is  4  hr.  53  min.  h^\  sec.  earlier 
than  in  London ;  what  is  the  difference  of  longitude  between 
the  two  places  ? 

5.  The  time  in  Berlin  is  44  min.  14^  sec.  later  than  in 
Paris ;  what  is  the  difference  of  longitude  between  the  two 
places  ? 

6.  The  difference  of  longitude  between  two  places  is 
71°  4' ;  what  is  the  difference  of  time  ? 

7.  Find  the  difference  of  time  between  New  York  (long. 
74°  0'  3"  W.)  and  San  Francisco  (long.  122°  25'  40"  W.). 

8.  Find  the  difference  of  time  between  Ottawa  (long. 
75°  42'  4"  W.)  and  Washington  (long.  77°  2'  48"  W.). 

9.  Find  the  difference  of  time  between  Bombay  (long. 
72°  54'  E.)  and  Cape  of  Good  Hope  (long.  18°  29'  E.). 

10.  Find  the  difference  of  time  between  Constantinople 
(long.  28°  59'  14"  E.)  and  Quebec  (long.  71°  13'  45"  W.). 

11.  Find  the  difference  of  time  between  Canton  (long. 
113°  14'  E.)  and  Chicago  (long.  87°  37'  30"  W.). 

12.  Find  the  difference  of  time  between  Boston  (long.  71° 
3'  30"  W.)  and  St.  Paul  (long.  93°  5'  W.). 

13.  Find  the  difference  of  time  between  Pekin  (long.  116° 
27'  E.)  and  New  York  (long.  74°  0'  3"  W.) 


154  ARITHMETIC. 

14.  Find  the  difference  of  time  between  Rome  (long.  12° 
28'  40"  E.)  and  Paris  (long.  2°  20'  14"  E.). 

102.  The  earth  revolves  on  its  axis  from  west  to  east, 
and  the  sun  seems  to  move  from  east  to  west.  Of  any  two 
places,  the  sun  rises  earlier  at  the  place  farther  east,  and 
since  the  sun  rises  earlier,  the  clock-time  is  later.  Hence, 
to  find  the  clock-time  of  a  given  place  when  the  clock-time 
of  another  place  and  their  difference  of  time  are  known, 
add  the  difference  of  time  to  the  given  time,  when  the  place 
whose  time  is  to  be  found  is  farther  east;  subtract  the  differ- 
ence of  time  from  the  given  time,  when  the  place  whose  time  is 
to  be  found  is  farther  west. 

I.  When  it  is  12  min.  30  sec.  past  2  p.m.  at  Berlin  (long. 
13°  23'  45"  E.),  what  is  the  time  at  New  York  (long.  74° 
y)   o     W . )  .  "Phg  difference  of  time  is  found 

13°  23'  45"  to  be  5  hr.  49  min.  35i  sec.    Since 

74°     0'     3"  New  York  is  west  of  Berlin,  the 

15)87°  23'  48"  ti™^  is  earlier;    hence  we  sub- 

5  hr.  49  min.  354-  sec.  *^^^*  ^  ^^-  ^^  ™^"'  ^^^  '^^-  ^^°"^ 

^  2  hr.  12  min.  30  sec.  p.m.    2  hr. 

2  hr.  12  min.  30  sec.  p.m.  ^^^^"^  "°°"  '^  ^^'^  ^^"^^  as  14  hr. 

5  49  35^  after  midnight,  and  as  we  can- 

o  1,      oo      •       KAA  not  subtract  5  hr.  from  2  hr.,  we 

8  hr.  22  mm.  54i  sec.  a.m.  ,  .  ^       , ,  ,  ' . 

^  subtract  it  from  14  hr.,  and  write 

Ans.  22  min.  54|-  sec.  past  8  a.m.    a.m.  instead  of  p.m. 

II.  What  is  the  longitude  of  a  place  whose  time  is  48 
min.  past  8  p.m.,  when  it  is  half  past  6  p.m.  at  Eome  (long. 
12°  28'  40"  E.)  ? 

8  hr.  48  min. 


The  difference  of  longitude  is  found  to  be  34° 
30^  Since  the  time  is  later,  the  place  is  east  of 
Rome  ;  hence  we  add  34°  30'  to  12°  28'  40". 


6 

30 

2hr, 

.  18  min. 
15 

34° 
12° 

30' 
28'  40" 

46°    58'40"E. 


SPECIAL  PROBLEMS.  165 

EXAMPLES. 

1.  Bangor  is  15°  39'  east  of  Cincinnati ;  what  time  is  it 
at  Bangor  when  it  is  5  o'clock  p.m.  at  Cincinnati  ? 

2.  The  longitude  of  Berlin  is  13°  23'  45"  E. ;  what  time 
is  it  at  Greenwich  when  it  is  midnight  at  Berlin  ? 

3.  What  is  the  time  at  Canton  (long.  113°  14'  E.)  when 
it  is  noon  at  Greenwich  ? 

4.  The  longitude  of  Boston  is  71°  3'  30"  W.,  and  of 
Paris  2°  20'  14"  E. ;  when  it  is  10  o'clock  a.m.  at  Boston, 
what  time  is  it  at  Paris  ? 

5.  The  longitude  of  St.  Petersburg  is  30°  19'  E.,  and  of 
New  York  74°  0'  3"  W. ;  when  it  is  1  p'elock  p.m.  at  St. 
Petersburg,  what  time  is  it  at  New  York  ? 

6.  When  it  is  10  o'clock  at  Boston,  what  time  is  it  at 
Amherst,  the  longitude  of  Boston  being  71°  3'  30"  W.,  and 
that  of  Amherst  being  72°  31'  50"  W.  ? 

7.  The  longitude  of  Boston  is  71°  3'  30"  W.,  and  that  of 
San  Erancisco  is  122°  25'  40"  W.  When  it  is  noon  at  Bos- 
ton, what  is  the  time  at  San  Francisco  ? 

8.  Eind  what  time  it  is  at  Cape  of  Good  Hope  (long.  18° 
29'  E.)  when  it  is  noon  at  St.  Paul  (long.  93°  5'  W.). 

9.  When  it  is  6  min.  15  sec.  past  4  a.m.  at  Pekin  (long. 
116°  27'  E.),  what  is  the  time  at  London  (long.  5'  48"  W.). 

10.  What  is  the  longitude  of  a  place  whose  time  is  42 
min.  42  sec.  past  8  p.m.  when  it  is  midnight  at  Greenwich  ? 

11.  What  is  the  longitude  of  a  place  who§e  time  is  6 
o'clock  A.M.  when  it  is  quarter  past  4  a.m.  at  Washington 
(long.  77°  2'  48"  W.)  ? 

12.  What  is  the  longitude  of  a  place  whose  time  is  35 
min.  past  10  a.m.  when  it  is.  5  o'clock  p.m.  at  Paris   (long. 


156  ARITHMETIC. 

13.  When  it  is  noon  at  St.  Paul  (long.  93°  5'  W.),  it  is 
37  min.  12  sec.  past  1  p.m.  at  Bangor ;  what  is  the  longitude 
of  Bangor  ? 

14.  When  it  is  9  o'clock  p.m.  at  Calcutta  (long.  88°  20'  E.), 
it  is  27  min.  19|-  sec.  past  5  p.m.  at  Jerusalem ;  what  is  the 
longitude  of  Jerusalem  ? 

Note.  The  time  considered  in  the  preceding  problems  is  the  actual 
local  time,  but  nearly  all  railroads,  cities,  and  towns  of  the  United 
States  now  use  standard  time,  which  is  the  time  of  some  particular 
meridian.  The  meridians  selected  are  those  which  are  respectively 
75°,  90°,  105°,  and  120°  west  of  Greenwich.  The  time  of  the  meridian 
75°  W.  is  known  as  Eastern  standard  time ;  that  of  90°  W.  is  Central 
standard  time;  that  of  105°  W.  is  Mountain  standard  time;  and  that 
of  120^  W.  is  Pacific  standard  time.  By  this  method,  when  there  is 
any  difference  of  time  between  two  places,  the  difference  is  one,  two, 
or  three  hours,  and  all  confusion  arising  from  different  local  times  is 
thereby  avoided. 


RATIO   AND   PROPORTION.  157 


CHAPTER  VIII. 

RATIO  AND  PROPORTION. 

Ratio. 

103.  The  relation  between  two  numbers  is  called  their 
ratio,  and  it  is  determined  by  dividing  the  first  by  the 
second.  The  sign  of  ratio  is  the  colon  (:),  which  is  the 
sign  of  division  with  the  line  omitted.  For  example,  6 : 4 
is  read  the  ratio  of  6  to  .4,  or  6  is  to  4,  and  its  value  is  6  -h  4, 
or  f. 

The  two  numbers  whose  values  are  compared  are  called 
the  terms  of  the  ratio,  and  together  they  form  a  couplet. 
The  first  term  is  called  the  antecedent,  and  the  second  term 
is  called  the  consequent. 

A  ratio  can  exist  between  two  concrete  numbers  only 
when  they  are  expressed  in  terms  of  the  same  unit,  and  the 
ratio  is  equal  to  the  ratio  of  the  corresponding  abstract 
numbers.     For  example,  8  pt. :  15  pt.  equals  8  :  15. 

When  each  term  of  a  ratio  is  a  single  number,  it  is  called 
a  simple  ratio.  The  product  of  two  or  more  simple  ratios 
is  called  a  compound  ratio.  A  simple  ratio  having  a  fraction 
in  either  term  is  also  called  a  complex  ratio.  For  example, 
8 :  13  and  2^ :  f  are  simple  ratios,  the  latter  of  which  is 

complex ;  q  !  i  o  r  is  a  compound  ratio. 

Since  antecedent  and  consequent  bear  the  same  relation 
to  each  other  as  dividend  and  divisor,  both  terms  of  a  ratio 
may  be  multiplied  or  divided  by  the  same  number  without 
affecting  the  value  of  the  ratio.     A  complex  ratio  can  be 


158  ARITHMETIC. 

simplified  by  multiplying  both  terms  by  their  least  common 
denominator. 

A  compound  ratio  can  be  simplified  by  taking  the  product 
of  the  antecedents  for  a  new  antecedent,  and  the  product  of 
the  consequents  for  a  new  consequent. 

When  the  antecedent  and  consequent  of  a  ratio  are  inter- 
changed, the  resulting  ratio  is  called  the  inverse  of  the 
given  ratio. 

I.    Eeduce  2J  :  5J  to  a  simple  ratio. 

^i  *  ^¥-  Multiplying  both  terms  by  12,  we  obtain  28  :  63 ;  and 

28  :  63.        then   dividing  both   terms   by  7,  we   obtain  4  :  9   as   the 
4  •    9         simplest  value. 


II.    Which  is  the  greater  ratio,  7:8  or  8:9? 


Expressing  the  ratios  in  a  fractional  form, 

^^'  we  have  |  and  *,  which,  after  reducing  to  their 

9  =  1^  =  YY'  least  common  denominator,  equal  f f  and  f f . 


8  =  l  =  ff- 

Q  _  8  —  64 

9  is  the  larger. 


than  7 


EXAMPLES. 


1.  Eeduce  3f  :  5^  to  a  simple  ratio. 

12 

2.  Eeduce  74 :  —  to  a  simple  ratio. 

3  •  7  ) 

3.  Eeduce  ^^ '  q  >■  to  a  simple  ratio. 

4.  Eeduce  of'.al  [-  to  a  simple  ratio. 

5.  Find  the  ratio  of  2  pk.  to  3  qt. 

6.  rind  the  ratio  of  4  gal.  to  1  cu.  ft. 

7.  Find  the  ratio  of  a  field  15  rd.  long  and  11  rd.  wide 
to  a  field  14  rd.  long  and  12  rd.  wide. 

8.  Which  is  the  greater  ratio,  7  :  11  or  8  :  12  ? 


RATIO   AND   PROPORTION.  159 

9.    "WnicK  is  the  greater  ratio,  ^  :  f  or  4| :  4|-  ? 
10.    Which  is  the  greater  ratio,  $2.50 :  |3.75  or  8  ft. :  12  ft.  ? 

Simple  Proportion. 

104.  All  expression  of  equality  between  the  two  ratios  is 
called  a  proportion,  and  the  four  terms  are  called  propor- 
tionals. When  a  proportion  consists  of  two  simple  ratios, 
it  is  called  a  simple  proportion. 

A  proportion  is  indicated  by  putting  a  double  colon  (: :) 
or  a  sign  of  equality  (  =  )  between  the  two  ratios.  For 
example,  4  :  6  : :  10  :  15  is  read  4  is  to  6  as  10  is  to  15;  4  :  6 
==  10  :  15  is  read  the  ratio  of  4  to  6  equals  the  ratio  of  10  to  15. 

The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes,  and  the  second  and  third  terms  are  called  the 
means. 

Three  numbers  are  said  to  be  in  proportion  when  the  ratio 
of  the  first  to  the  second  equals  the  ratio  of  the  second  to 
the  third.  The  second  number  is  called  a  mean  proportional 
between  the  other  two.  For  example,  in  the  proportion 
2  :  6  : :  6  :  18,  6  is  a  mean  proportional  between  2  and  18. 

The  solution  of  problems  in  proportion  depends  on  the 
following  principle  :  —  In  any  projyortion  the  product  of  the 
extremes  equals  the  product  of  the  means.  This  can  be  proved 
in  any  proportion  by  expressing  the  ratios  in  fractional 
form,  and  then  multiplying  both  of  them  by  the  product  of 
the  denominators.  As  an  illustration,  take  the  proportion 
2  :  3  :  :  4  :  6. 

2  :  3  :  :  4  :  6. 

2  _  4_  The  proportion  written  in  fractional  form 

becomes  |  =  |.     Multiplying  both  fractions 
2x3x6^4x3x^,  by  3x6,  the  results  are  still  equal.     Hence 

^  ^  2X6  =  4x3, 

2X6  =  4X3. 


160  .  ARITHMETIC. 

From  the  above  principle  it  follows  that  either  extreme 
equals  the  product  of  the  means  divided  by  the  other  extreme i 
and  either  mean  equals  the  product  of  the  extremes  divided  by 
the  other  mean. 

I.  Find  a  fourth  proportional  to  8,  10,  and  12. 

8  :  10  :  :  12  :  ic.  IJet  the  required  term  be  represented  by 

X.   Then  8  :  10  : :  12  :  a;.    Since  either  extreme 

equals  the  product  of  the  means  divided  by 

10  X 12 
the  other  extreme,  x— — — —  =  15. 

8 

II.  Find  the  second  term  of  a  proportion  of  which  the 
first,  third,  and  fourth  terms  are  respectively  2|,  3f ,  and  7^. 

2f  :  aj :  :  3f  :  71. 

Let  the  required  term  be  represented 

X  —  ^jXi^-^^J'  hy  X.     Then   2|  :  x  :  :  3f  :  71.      Since 

3  either  mean  equals  the  product  of  the 

_  X^  y^lE  y^  J_ —  ^  —  ^1  extremes  divided  by  the  other  mean, 

^       2       ^1      4  :r  =  2fx7i-^3f=5i. 


EXAMPLES. 

1.  Find  a  fourth  proportional  to  9,  51,  and  75. 

2.  Find  the  third  term  of  a  proportion  of  which  the 
first,  second,  and  fourth  terms  are  respectively  18,  15,  and 
100. 

3.  Find  the  number  which  has  to  6f  the  same  ratio 
which  llf  has  to  3f 

4.  Find  the  number  to  which  8^  has  the  same  ratio 
which  25  has  to  37^. 

..'5.   Find  the  third  term  of  a  proportion  of  which  the 
first,  second,  and  fourth  terms  are  respectively  -^,  J,  and  ^ 

6,   Find  a  fourth  proportional  to  3.75,  0.23,  and  0,16. 


BATTO  AND  PROPORTION.  161 

7.  Find  the  number  which  has  to  0.649  the  same  ratio 
which  58  has  to  634. 

8.  Find  the  fourth  term  of  a  proportion  of  which  the 
first,  second,  and  third  terms  are  respectively  3.81,  0.056, 
and  1.67. 

105.  The  method  of  finding  either  term  of  a  proportion 
when  the  other  three  are  known  is  often  called  the  rule  of 
three.  It  is  customary  to  represent  the  required  term  by  x, 
and  then  arrange  the  terms  so  that  x  will  be  the  fourth 
term.  The  number  in  the  problem  which  corresponds  to 
the  answer  must  be  the  third  term. 

I.  If  15  yards  of  silk  cost  |36,  what  will  25  yards  cost? 

Since  the  answer  is  to  be  dollars,  make  S3G  the 
15  :  25  :  :  36  :  cc        third  terra.     25  yards   will   cost   more   than   16 
6      12 
23  x  M  yards,  and  the  fourth  term  will  be  greater  than 

^  ~-  — Y^  =  ^^^-      the  third ;  lience  the  second  term  must  be  greater 
^  than  the  first,  and  the  first  couplet  is  16  :  25.    The 

answer  is  then  found  as  in  the  preceding  section. 

II.  If  8  men  can  do  a  piece  of  work  in  5  days,  how  long 
will  it  take  10  men  to  do  the  same  work  ? 

10  :  8  :  :  5  :  a;  Since  the  answer  is  to  be  days,  make  6  days  the 

4  third  terra.     10  men  can  do  the  work  in  less  time 

a;z=?il^=4(lays.  *^^"  ^  "^^"'  ^"^  *'^^  fourth  term  will  be  smaller 

^^  than  the  third ;  hence  the  second  term  must  be 

^  smaller  than  the  first,  and  the  first  couplet  is  10  : 8. 

The  answer  is  then  found  as  in  the  preceding  section. 

In  the  solution  of  problems  in  simple  proportion,  make 
that  mimber  the  third  term  ivhich  is  of  the  same  kind  as  the 
required  answer.  If  from  the  nature  of  the  question  the 
answer  is  to  be  greater  than  the  third  term,  make  the  greater 
of  the  other  two  numbers  the  second  term,  and  the  smaller  the 
first;  if  the  answer  is  to  be  smaller  than  the  third  term,  make 


162  ARTTHMEtlC. 

the  second  term  smaller  than  the  first.  Divide  the  product  of 
the  means  by  the  first  term,  and  the  quotient  is  the  fourth  term, 
or  answer. 

EXAMPLES. 

1.  If  18  barrels  of  flour  last  a  garrison  8  weeks,  how 
long  will  63  barrels  last  ? 

2.  If  the  rent  of  36"*  of  land  is  $48,  how  many  hektars* 
can  be  rented  for  $84  ? 

3.  If  a  stock  of  provisions  will  supply  a  garrison  of  240 
men  96  days,  how  long  will  the  same  stock  supply  384  men  ? 

4.  If  36  men  can  do  a  piece  of  work  in  22  days,  how 
many  men  can  do  the  same  work  in  8  days  ? 

5.  If  a  clock  ticks  120  times  in  a  minute,  how  many 
times  will  it  tick  in  2^  hours  ? 

6.  If  3  lb.  7  oz.  of  butter  cost  $1.10,  what  will  14f  lb. 
cost? 

7.  If  a  train  runs  160^"'  in  3  hours,  how  long  will  it 
take  it  to  run  70«^'"  ? 

8.  If  a  man  earns  $16  in  5  days,  how  much  will  he  earn 
in  14  days  ? 

9.  If  22  yd.  of  silk  18  in.  wide  are  required  for  a  dress, 
how  many  yards  of  cloth  30  in.  wide  would  be  required  for 
a  similar  dress  ? 

10.  A  field  can  be  mowed  in  4  days  of  11  hours  each ; 
how  many  days  of  9  hours  each  will  it  take  ? 

11.  If  12  men  can  build  a  wall  19  rods  long  in  a  day,  how 
long  a  wall  will  32  men  build  in  the  same  time  ? 

12.  If  14  yd.  of  cloth  32  in.  wide  will  make  a  dress,  how 
many  yards  of  cambric  24  in.  wide  will  be  required  to  line 
it? 


RATIO   AND   PROPORTION.  163 

13.  If  10.5"'  of  wood  cost  ^12.25,  how  many  sters  can  be 
bought  for  f  50  ? 

14.  If  f  of  a  warehouse  is  worth  $7000,  what  is  ^  of  it 
worth  ? 

15.  If  a  post  5  ft.  4  in.  high  casts  a  shadow  6  ft.  4  in. 
long,  how  long  a  shadow  wiH  be  cast  by  a  steeple  176  ft. 
high? 

16.  At  the  time  when  a  man  5  ft.  9  in.  in  height  casts  a 
shadow  4  ft.  6  in.  long,  what  is  the  height  of  a  tree  that 
casts  a  shadow  52  ft.  6  in.  long  ? 

17.  If  a  cistern  can  be  filled  in  2  hr.  27  min.  by  3  pipes, 
in  what  time  can  it  be  filled  by  7  pipes  of  the  same  size  ? 

18.  If  9J  yards  of  cloth  cost  $23|,  how  many  yards  can 
be  bought  for  |38-/^  ? 

19.  What  is  the  cost  of  60.5  tons  of  coal  when  0.9  of  a 
ton  costs  ^6.66  ? 

20.  A  merchant  failed  and  paid  60  cents  on  a  dollar; 
how  much  would  a  creditor  receive  whose  bill  Avas  $1426  ? 

21.  If  108^1  of  oats  last  100  horses  9  days,  how  long  will 
192H'  last  them  ? 

22.  If  a  man  travels  64  rods  in  0.05  of  an  hour,  how 
many  minutes  will  it  take  him  to  go  a  mile  ? 

23.  If  18  men  can  perform  a  piece  of  work  in  42  days,  in 
how  many  days  can  they  perform  the  same  work  with  the 
assistance  of  9  more  men  ? 

24.  A  piece  of  work  can  be  done  in  50  days  by  35  men. 
After  12  days  16  men  strike.  In  how  many  days  will  the 
rest  finish  the  work  ? 

25.  If  6iT.  of  coal  cost  £6  15  s.  5d.,  what  will  be  the 
price  of  5  T.  3  cwt.  ? 


164  ARITHMETIC. 

26.  By  a  pipe  of  a  certain  capacity  a  cistern  can  be 
emptied  in  3^^  hours ;  in  what  time  can  it  be  emptied  by 
a  pipe,  the  capacity  of  which  is  |  greater  ? 


Compound  Proportion. 

106.  An  expression  of  equality  between  a  compound 
ratio  and  a  simple  ratio,  or  between  two  compound  ratios, 
is  called  a  compound  proportion. 

I.  If  4  men  dig  a  trench  84  feet  long  and  5  feet  wide  in 
3  days  of  8  hours  each,  how  many  men  can  dig  a  trench 
420  feet  long  and  3  feet  wide  in  4  days  of  9  hours  each  ? 


84 :  420 
5:3 
4:3 
9:8 


Since   the   answer  is  to  be  men, 

make  4  men  the  third  term.     The 

:  :  4  :  07.  number  of   men  required   depends 

upon   four  conditions  —  the   length 

of    the    trench,   the    width   of    the 

^^      n     o     Q      *  trench,  the  number  of  days,  and  the 

^^^^^X^X>^X8X^^g  men.      number  of   hours   per   day;    all  of 

'^  ^  these  must  be  considered  m  statmg 

the  problem.     A  trench  420  ft.  long 

will  require  more  men  than  a  trench  84  ft.  long,  and  the  first  ratio  is 

84 :  420 ;  a  trench  3  ft.  wide  will  require  less  men  than  a  trench  5  ft, 

wide,  and  the  second  ratio  is  5:3;  to  complete  the  work  in  4  da.  will 

require  less  men  than  to  complete  it  in  3  da.,  and  the  third  ratio  is 

4:3;  days  of  9  hr.  each  will  require  less  men  than  days  of  8  hr.  each, 

and  the  fourth  ratio  is  9:8.     Dividing  the  product  of  the  means  by 

the  product  of  the  given  extremes,  we  have  8  men  as  the  answer. 

In  the  solution  of  problems  in  compound  proportion, 
make  that  number  the  third  term  which  is  of  the  same  kind  as 
the  required  answer.  Take  the  other  terms  in  pairs  of  the 
same  kind,  and  form  a  ratio  of  each  pair  as  in  simple  propor- 
tion. Divide  the  product  of  the  means  by  the  product  of  the 
given  extremes,  arid  the  quotient  is  the  fourth  term,  or  answer. 


RATIO  AUD  PROPORTION.  165 


EXAMPLES. 

1.  If  6  men  in  15  days  earn  $135,  how  much  will  9 
men  earn  in  18  days  ? 

2.  If  6  men  can  dig  6  rods  of  a  ditch  in  6  hours,  how; 
many  rods  will  12  men  dig  in  12  hours  ? 

3.  If  16  men  build  18  rods  of  wall  in  12  days,  how 
many  men  will  be  needed  to  build  72  rods  in  8  days  ? 

4.  If  the  wages  of  12  men  for  8  days  of  8  hours  each 
are  f  135,  what  will  be  the  wages  of  25  men  for  12  days  of 
10  hours  each  ? 

5.  If  a  man  travels  117  miles  in  15  days,  travelling  9 
hours  a  day,  how  far  would  he  go  in  20  days,  travelling  12 
hours  a  day  ? 

6.  If  5  men  can  do  a  piece  of  work  in  7  days  of  10  hours 
each,  in  how  many  days  can  12  men  do  the  same,  working 
8  hours  per  day  ? 

7.  If  2|-  acres  of  pasturage  can  support  5  oxen  for  3^ 
days,  how  many  acres  would  be  required  to  support  26  oxen 
for  17-^  days  ? 

8.  If  14  horses  eat  70  bushels  of  grain  in  20  days,  how 
many  bushels  will  suffice  30  horses  50  days  ? 

9.  If  8  horses  consume  3|  tons  of  hay  in  30  days,  how 
long  will  4^  tons  last  10  horses  ? 

10.  If  9  men  build  247y23  rods  of  wall  in  28  days,  in  how 
many  days  will  8  men  build  51  rods  ? 

11.  If  2  men,  working  8  hours,  can  carry  12000  bricks  to 
the  height  of  50  feet,  how  many  bricks  can  1  man,  working 
10  hours,  carry  to  the  height  of  30  feet  ? 


166  ARITHMETIC. 

12.  If  a  six  cent  loaf  weighs  8  ounces  when  wheat  is 
$1.25  per  bushel,  how  much  bread  may  be  bought  for  50 
cents  when  wheat  is  f  1.00  per  bushel  ? 

13.  If  49  men  can  empty  a  reservoir  in  65  days,  pumping- 
8  hours  a  day,  how  many  hours  a  day  must  196  men  pump 
to  empty  it  in  26  days  ? 

14.  If  5  horses  will  consume  8  bu.  1  pk.  6  qt.  of  oats  in 
6  days,  what  quantity  of  oats  will  7  horses  consume  in  11 
days  ? 

15.  If  it  take  35^^  of  wool  to  make  a  piece  of  cloth  25™ 
long  and  f ""  wide,  how  long  a  piece  of  cloth,  i™  wide,  can  be 
made  from  112^«  ? 

16.  If  a  family  of  9  persons  spends  $305  in  4  months, 
how  many  dollars  will  maintain  it  8  months,  if  5  persons 
were  added  to  the  family  ? 

17.  If  a  man  travels  1440  miles  in  36  days,  travelling  10 
hours  a  day  at  the  rate  of  4  miles  an  hour,  in  what  time 
will  he  travel  576  miles,  going  8  hours  a  day  at  the  rate  of 
3  miles  per  hour  ? 

18.  If  3  men  can  build  a  wall  60  feet  long,  8  feet  high, 
and  3  feet  thick,  in  64  days  of  9  hours,  how  many  days  of 
8  hours  will  20  men  require'  to  build  a  wall  400  feet  long,  9 
feet  high,  and  5  feet  thick  ? 

19.  If  a  slab  of  marble,  8  feet  long,  3  feet  wide,  and  3 
inches  thick,  weighs  1050  pounds,  how  much  will  another 
slab  of  the  same  marble  weigh  which  is  6  feet  long,  2  feet 
wide,  and  2  inches  thick  ? 

20.  If  6  iron  bars,  4  feet  long,  3  inches  broad,  and  2 
inches  thick,  weigh  288  pounds,  find  the  weight  of  15  bars, 
each  6^  feet  long,  2^  inches  broad,  and  1^  inches  thick. 


KATIO   AND  PROPORTION.  167 

21.  If  25  men,  working  8  hours  a  day,  do  f  of  a  piece  of 
work  in  24  days,  in  liow  many  days  of  10  hours  each  will 
30  men  finish  the  piece  of  work  ? 

22.  If  12  pipes,  each  delivering  12  gallons  a  minute,  fill 
a  cistern  in  3  hr.  24  niin.,  how  many  pipes,  each  delivering 
16  gallons  a  minute,  will  fill  a  cistern  6  times  as  large  in 
6  hr.  48  min.  ? 

23.  A  man  has  a  bin  7  ft.  long,  2^  ft.  wide,  and  2  ft. 
deep,  which  contains  28  bushels  of  corn ;  how  deep  must  he 
build  another,  which  is  to  be  18  ft.  long,  1  ft.  10^  in.  wide, 
in  order  to  contain  120  bushels  ? 

24.  If  496  men,  in  5  days  of  12  hr.  6  min.  each,  dig  a 
trench  of  9  degrees  of  hardness,  465  ft.  long,  3J  ft.  wide, 
and  4|  ft.  deep,  how  many  men  will  be  required  to  dig  a 
trench  of  2  degrees  of  hardness,  168|  ft.  long,  7J  ft.  wide, 
and  21  ft.  deep,  in  22  days  of  9  hr.  each  ? 

Cause  and  Effect. 

107.  Problems  in  proportion  can  also  be  solved  by  the 

application  of  the  following  principle :  Like  causes  produce 
like  effects,  and  the  ratio  hetioeen  any  two  causes  equals  the 
ratio  between  the  effects  produced. 

Note.  As  examples  of  causes  may  be  mentioned  men  at  work,  time, 
and  goods  bought  or  sold ;  as  examples  of  effects,  work  done,  wages, 
and  cost  of  goods. 

I.   If  15  yards  of  silk  cost  f  36,  what  will  25  jards  cost? 

15  :  25  :  :  36  :  iC  Let  the  required  number  of  dollars  be  rep- 

6      12  resented  by  x.     The  first  and  se  lond   causes 

/gsa^iilif  =$60.  ^re  respectively  15  yd.  and  25  yd      The  first 

'^  and  second  effects  are  respectively  $36  and  x 

dollars.    Hence  the  proportion  is  15:  26::  36:  x 


168  ARITHMETIC. 

II.  If  8  men  can  do  a  piece  of  work  in  5  days,  how  long 
will  it  take  10  men  to  do  the  same  work  ? 

8  :  10  )       ^     ^  Let  the  required  number  of  days  be  repre- 

5  :  ic    j  ■  *      ■  sented  by  x.     The  first  causes  are  8  men  and 

4  5  days,  and  the  second  causes  are  10  men  and 

g._.^X^Xl_.^  days.  ^  ^^ys-     The   effects   are   the  same,  and  can 

-^^Xl  each  be  represented  by  1.     Hence  the  i)ro- 

portion  is    r\       V  : :  1 : 1.     Since  x  is  a  mean, 

its  value  is  found  by  dividing  the  product  of  the  extremes  by  the 

product  of  the  given  means. 

III.  If  4  men  dig  a  trench  84  feet  long  and  5  feet  wide 
in  3  days  of  8  hours  each,  how  many  men  can  dig  a  trench 
420  feet  long  and  3  feet  wide  in  4  days  of  9  hours  each  ? 

4:ic)         (qj     ir)A  "^^^  *''^  required  number  of  men 

3  :  4  >-  : :  -<     k    o  ^^  represented  by  x.  The  first  causes 

8:9)         I    ^  •  '^  are  4  men,  3  da.,  and  8  hr.,  and  the 

^^  second  causes  are  x  men,  4  da.,  and 

^^^X3x8x^^Px3_g  ^^^     9hr.     The  first   effect  is  a  trench 

^XgX^^X^  '84  ft.  long  and  5  ft.  wide,  and  the 

second  effect  is  a  trench  420  ft,  long 

and  3  ft.  wide.    Hence  the  proportion  is  3:41::-^    ^io     • 

Note.  The  illustrative  problems  are  the  same  three  that  were  ex- 
plained by  the  rule  of  three  in  the  two  preceding  sections.  All  prob- 
lems given  under  either  head  can  be  solved  by  either  method. 

EXAMPLES. 

1.  How  many  hektars  of  land  can  be  bought  for  $84, 
when  3^*  can  be  bought  for  $26.25? 

2.  If  a  horse-car  goes  4  miles  in  35  minutes,  how  far 
will  it  go  in  3  hours  ? 

3.  If  a  tree  24  feet  high  casts  a  shadow  30  feet  long, 
what  must  be  the  height  of  a  building  to  cast  a  shadow  55 
feet  long  ? 


BATIO  AND  PROPORTION.  169 

4.  If  16™  of  silk  cost  120  francs,  what  will  25"  cost  ? 

5.  If  14  men  can  build  a  wall  in  10  days,  how  many 
men  will  it  take  to  build  the  same  wall  in  7  days  ? 

6.  If  67.5"  of  carpeting  80*="*  wide  will  cover  a  floor,  how 
many  meters  90*""  wide  will  it  take  to  cover  it  ? 

7.  If  a  pasture  of  18  acres  will  feed  8  cows  5  months, 
how  many  months  will  a  pasture  of  27  acres  feed  12  cows  ? 

8.  A  man  receives  f  18  for  6  days'  work  of  8  hours 
each ;  what  should  he  receive  for  5  days'  work  of  9  hours 
each? 

9.  If  8  men  spend  ^32  in  13  weeks,  what  will  24  men 
spend  in  52  weeks  ? 

10.  If  the  wages  of  72  men  for  5  days  is  f 450,  how 
many  men  may  be  hired  for  12  days  for  $540  ? 

11.  A  man,  travelling  9  hours  a  day,  goes  234  miles  in 
15  days ;  how  far  can  he  go  in  30  days,  travelling  8  hours  a 
day? 

12.  If  a  man  travelling  uniformly,  7  hours  per  day,  goes 
455  miles  in  26  days,  how  far  can  he  go  in  20  days,  travel- 
ling 9  hours  per  day  at  the  same  rate  per  hour  as  before  ? 

13.  If  6  men  can  build  20  feet  of  a  stone  wall  in  10  days, 
how  many  men  can  build  360  feet  of  the  same  wall  in  90 
days  ? 

14.  If  17  men  can  reap  a  field  in  9  days,  how  long  would 
it  take  to  reap  half  of  it  when  5  men  refuse  to  work  ? 

15.  If  3  men  can  reap  8  acres  in  5  days,  working  8  hours 
a  day,  in  how  many  days  can  8  men,  working  12  hours  a 
day,  reap  192  acres  ? 

16.  If  it  costs  f  7.20  to  transport  18^  cwt.  5-^  miles,  what 
will  it  cost  to  transport  112f  tons  62^  miles  ? 


170  ARITHMETIC. 

17.  If  27  men,  working  10  hours  a  day,  do  a  piece  of 
work  in  14  days,  how  many  hours  a  day  must  12  men  work 
to  do  the  same  amount  of  work  in  45  days  ? 

18.  If  24  men  can  saw  90  cords  of  wood  in  6  days,  when 
the  days  are  9  hours  long,  how  many  cords  can  8  men  saw 
in  36  days,  when  they  are  12  hours  long  ? 

19.  If  a  block  of  granite  8  ft.  long,  2  ft.  wide,  and  1  ft., 
6  in.  thick,  weighs  920  lb.,  how  much  will  a  block  of  the 
same  kind  of  granite  weigh  which  is  12  ft.  long,  3  ft.  wide, 
and  2  ft.  thick  ? 

20.  If  6  men  do  a  certain  piece  of  work  in  17  days  of  9 
hours  each,  how  many  days  of  8|-  hours  each  will  24  men, 
working  at  the  same  rate,  require  to  do  20  such  pieces  ? 

21.  A  wall  which  was  to  be  36  ft.  high  was  raised  9  ft. 
in  16  days  by  16  men ;  how  many  men  will  be  needed  to 
finish  the  work  in  4  days  ? 

22.  If  8  ounces  of  bread  can  be  bought  for  10  cents  when 
wheat  is  $1.00  per  bushel,  what  weight  of  it  may  be  bought 
for  18  cents  when  the  price  of  wheat  is  $1.12  per  bushel  ? 

23.  If  30  lb.  of  cotton  will  make  3  pieces  of  muslin  42 
yd.  long  and  f  yd.  wide,  how  many  pounds  will  it  take  to 
make  50  pieces,  each  containing  35  yd.,  1-J  yd.  wide  ? 

24.  If  6  men  can  build  a  wall  80  ft.  long,  10  ft.  high,  and 
9  ft.  thick  in  100  days  of  9  hours,  how  many  days  of  10 
hours  will  be  required  by  15  men  to  build  a  wall  200  ft. 
long,  9  ft.  high,  and  5  ft.  thick  ? 

25.  If  5  compositoi-s  in  16  days,  11  hours  long,  can  com- 
pose 25  sheets  of  24  pages  in  each  sheet,  44  lines  in  a  page, 
and  40  letters  in  a  line,  in  how  many  days,  10  hours  long, 
can  9  compositors  compose  a  volume  (to  be  printed  in  the 
same  kind  of  type),  consisting  of  36  sheets,  16  pages  to  a 
sheet,  50  lines  to  a  page,  and  45  letters  to  a  line  ? 


RATIO   AND  PROPORTION.  171 


Partitive  Proportion. 

108.  The  process  of  dividing  a  number  into  parts  which 
are  proportional  to  given  numbers  is  called  partitive  pro- 
portion, and  the  parts  are  called  proportional  parts. 

I.   Divide  168  into  four  parts  which  shall  be  to  each  other 

as  3,  5,  7,  and  9. 

3  +  5  +  7  +  9  =  24 

—  of  ^^^  =  21;  The  number  168  may  be   conceived   as 
^^                                    divided  into  a  number  of  equal  parts,  3  of 

r  7  wliicli  make  up  the  first  part,  5  the  second, 

—  of  X^^  =  35 ;  7  the  third,  and  9  the  fourth ;  thus  the  num- 

ber of  equal  parts  is  24.     Hence  the  first 

—  of  108  — 49'  P^^*  equals  -^^  of  168,  or  21;    the  second 
^^                      '  part  equals  ,\  of  168,  or  35 ;  the  third  part 

7  equals  ^j  of  168,  or  49;    and  the  fourth 

^  of  X^^  =  63.  part  equals  ^%  of  168,  or  63. 

^ns.  21, 35, 49,  and  63. 


II.   Divide  $580  into  three  parts  which  shall  be  to  each 
other  as  ^,  f,  and  1\. 

1=3-6^    6+8  +  15  =  29 

2Q  Reducing     the    fractions  to   their 

l=T%     A  of  ^^p  =  120 ;  L.C.D.,    we    have    j%,   j%,    and    |f. 

^  J J  5      r9  Since    the    fractions    now  liave   the 

*~12                  20  same  denominator,  the   parts  which 

—  of  ,*^P=1dO;  are  proportional  to  the  fractions  are 

proportional  to  the  numerators.  Hence 

—  of  S80=r3OO  ^^  divide  $580  into  three  parts  pro. 
^9                 '      '  portional  to  6,  8,  and  15. 

Ans.  $120,  $160,  and  $300. 


172  ARITHMETIC. 

EXAMPLES. 

1.  Divide  324  into  two  parts  which  shall  be  to  each 
other  as  19  to  8. 

2.  Divide  90  into  five  parts  which  shall  be  to  each  other 
as  1,  2,  S,  4,  and  5. 

3.  Divide  968  into  three  parts  which  shall  be  to  each 
other  as  2|-,  3J,  and  4J. 

4.  Divide  420  into  three  parts,  such  that  they  shall  be 
proportional  to  ^,  |,  and  J. 

5.  Divide  the  reciprocal  of  8  into  two  parts  which  shall 
be  to  each  other  as  the  reciprocals  of  4  and  2|. 

6.  Coffee  is  mixed  in  the  ratio  of  2  lb.  of  Java  to  1  lb. 
of  Mocha;  how  much  of  each  kind  is  there  in  a  mixture 
weighing  75  lb.  ? 

7.  The  cost  of  a  horse  and  harness  was  $384,  and  the 
horse  cost  seven  times  as  much  as  the  harness ;  find  the 
cost  of  each. 

8.  An  alloy  contains  325  parts  of  copper  to  175  parts  of 
zinc ;  how  much  of  each  metal  is  contained  in  43^^  850^  of 
this  alloy? 

9.  If  bell  metal  is  made  of  25  parts  of  copper  to  11 
parts  of  tin,  find  the  weight  of  each  metal  in  a  bell  weigh- 
ing 1044  pounds. 

10.  A  father  divided  f  1550  among  three  sons  in  parts 
proportional  to  their  ages,  which  were  respectively  17,  20, 
and  25  years ;  how  much  did  each  receive  ? 

11.  A  man  said,  "  I  will  spend  half  my  income,  save  a 
third  of  it,  and  devote  a  fourth  to  business."  His  income 
was  $780.  Point  out  his  blunder,  and  divide  his  income 
rightly  in  the  proportion  intended  by  him. 


RATIO  AND  PROPORTION.  178 

12.  Gunpowder  is  composed  of  nitre,  charcoal,  and  sul- 
phur, in  the  proportion  of  15,  3,  and  2.  A  certain  quantity 
of  gunpowder  is  known  to  contain  20  cwt.  of  charcoal ;  find 
its  weight,  and  also  the  weight  of  nitre  and  sidphur  it 
contains. 

Simple  Partnership. 

109.  An  association  of  two  or  more  persons  for  the 
transaction  of  business  is  called  a  partnership.  Such  a 
partnership  association  is  called  a  firm,  company,  or  house, 
and  the  persons  associated  together  are  calletl  partners. 
The  money  and  property  invested  in  the  business  is  called 
capital  or  stock.  The  resources  or  assets  of  a  firm  are  its 
property  of  all  kinds  together  with  the  amounts  due  it ;  the 
liabilities  of  a  firm  are  its  debts. 

When  the  capital  of  several  partners  is  invested  for  the 
same  time,  the  partnership  is  called  simple  partnership. 
The  profits  and  losses  are  shared  in  proportion  to  the  amount 
of  capital  each  partner  has  invested  in  the  business,  except 
when  some  other  special  agreement  has  been  made. 

I.  A,  B,  and  C  formed  a  partnership ;  A  put  in  ^700,  B 
^800,  and  C  f  1000 ;  what  was  each  partner's  share  of  a  profit 
f^ounting  to  |950  ? 


^-^of  m  =  ^QQ; 

700 

%m 

The  method  of    partitive  propor- 

800 

^^^.im=^^^'. 

tion  is  used,  dividing  the  profit  into 

1000 

parts  proportional  to  700,  800,  and 

2500 

1000. 

^  of  ^^0^380. 

%m 

A,  %2m',  B,  $304;  C,  $380. 


174  ARITHMETIC. 


EXAMPLES. 

1.  A  and  B  form  a  partnership,  A  putting  in  $3000  and 
B  $2500 ;  what  is  each  partner's  share  of  a  profit  amounting 
to  $2200? 

2.  A,  B,  and  C  invested  in  trade  as  follows :  A  $800, 
B  $600,  and  C  $900.  What  was  each  partner's  share  of  a 
profit  amounting  to  $1350? 

3.  A  and  B  formed  a  partnership,  and  A's  capital  was 
equal  to  J  of  B's ;  what  was  each  partner's  share  of  a  profit 
amounting  to  $3600  ? 

4.  A,  B,  C,  and  D  traded  in  company.  A  put  in  $7500, 
B  $7000,  C  $9500,  and  D  $8000;  what  was  each  partner's 
share  of  a  profit  amounting  to  $9280  ? 

5.  A,  B,  and  C  formed  a  partnership,  A  putting  in  $1500, 
B  $1800,  and  C  $1400.  On  closing  business  they  found 
they  had  lost  $800.     What  was  the  loss  of  each? 

6.  A,  B,  and  C  hired  a  pasture  for  $100;  A  put  in  12 
cows,  B  8  cows,  and  C  5  cows ;  how  much  should  each  pay  ? 

7.  A  bankrupt  owed  $550  to  A,  $675  to  B,  and  $875  to 
C.  His  entire  property  was  sold  for  $1043.28 ;  what  was 
each  creditor's  share? 

8.  A,  B,  and  C  engaged  in  trade  with  a  joint  capital  of 
$9000.  At  the  end  of  a  year  A's  gain  was  $1250,  B's  $1000^ 
and  C's  $1500.  What  was  each  partner's  share  of  the 
capital  ? 

Compound  Partnership. 

110.  When  the  capital  of  the  several  partners  is  invested 
for  unequal  times,  the  partnership  is  called  compound  part- 
nership.   The  division  of  the  profits  a»d  losses  depends  both 


RATIO   AND  PROPORTION.  175 

on  the  amount  of  each  partner^s  capital  and  the  time  for 
which  it  is  invested. 

I.  A,  B,  and  C  invested  in  trade  as  follows  :  A  $300  for 
10  months,  B  $400  for  8  months,  and  C  $600  for  6  months. 
What  was  each  partner's  share  of  a  profit  amounting  to 
11960? 

20  The  use  of  §300  for 

OAA    .  iA       OAAA      ^^  of  ^^^P  =  600  ;      lOiTio.   is   equivalent   to 

300x10  =  3000     Qmo       ^^  ,,  ^  ,^,.       ^^f,,, 

400  X    8  =  3200     Zll       20  *^^7,l'"^^^\""''^T' 

600  X    6  =  3600     ^  ^^  ^^^^  =  ^^^ '     """  ^^    '   ""'     '"''• ' 

^^P0  use  of  .$400  for  8  mo.  is 

9800      3600    .  j^„„rt  _  Y20        equivalent  to  the  use  of 


8   times  $400,  or  $3200, 
A,  f  600 ;  B,  $640 ;  C,  f  720.  ^'„„\:V„:";  Z^. 

alent  to  the  use  of  6  times  $600,  or  $3600,  for  1  mo.  The  amounts 
invested  are  thus  reduced  to  the  same  standard,  and  the  profit  is 
divided  into  parts  proportional  to  3000,  3200,  and  3600. 


EXAMPLES. 

1.  A  and  B  enter  into  partnership.  A  contributes 
$1200  for  13  months,  and  B  $1600  for  10  months.  What 
is  the  share  of  each  in  a  gain  of  $1300? 

2.  Three  partners.  A,  B,  and  C,  furnish  capital  as  fol- 
lows :  A  $500  for  2  months,  B  $400  for  3  months,  and  C 
$200  for  4  months.  They  gain  $600 ;  what  is  each  part- 
ner's share  ? 

3.  Two  men  hire  a  pasture  for  $50;  one  puts  in  20 
horses  for  12  weeks,  and  the  other  25  horses  for  10  weeks. 
How  much  should  each  pay  ? 

4.  A,  B,  and  C  hire  a  pasture  for  $92.  A  pastures  6 
horses  for  8  weeks,  B  12  oxen  for  10  weeks,  and  C  50  cows 
for  12  weeks.  If  5  cows  are  reckoned  as  3  oxen,  and  3  oxen 
as  2  horseS;  how  much  shall  each  man  pay  ? 


176  ARITHMETIC. 

5.  Three  men  harvested  and  thrashed  a  field  of  grain  on 
shares,  A  furnishing  4  hands  5  days,  B  6  hands  4  days,  and 
C  5  hands  8  days.  The  whole  crop  was  630  bushels,  of 
which  they  had  one  fifth ;  how  much  did  each  receive  ? 

6.  Three  men  contract  to  do  a  piece  of  work  for  $8775. 
The  first  man  employs  20  men,  24  days,  10  hours  a  day ; 
the  second  25  men,  20  days,  12  hours  a  day ;  the  third  30 
men,  25  days,  9  hours  a  day.  How  much  should  each  of 
the  contractors  receive  ? 

7.  A,  B,  and  C  contract  to  build  a  piece  of  railroad  for 
$7500.  A  employs  30  men  50  days  ;  B  employs  50  men  36 
days ;  and  C  employs  48  men  and  10  horses  45  days  (each 
horse  to  be  reckoned  equal  to  1  man),  and  is  to  have 
$115.50  for  overseeing  the  work.  How  much  is  each  man 
to  receive  ? 

8.  A  and  B  rent  a  pasture  for  $690  per  annum.  A  putL 
in  200  sheep,  and  B  160 ;  at  the  end  of  6  months  they  dis- 
pose of  half  their  stock  and  allow  C  to  put  in  120 ;  what 
should  A,  B,  and  C  pay  severally  towards  the  rent  at  the 
year's  end  ? 

9.  A  and  B  entered  into  partnership  for  one  year.  A 
had  $800  in  the  business  during  the  first  4  months,  and 
$400  more  during  the  remainder  of  the  year ;  B  had  $500 
during  the  first  7  months,  and  $1300  during  the  last  5 
months.  At  the  end  of  the  year  they  found  they  had  lost 
$3800 ;  what  was  each  partner's  loss  ? 

10.  A,  B,  and  C  formed  a  partnership  and  cleared  $1200. 
A  put  in  $8000  for  4  months,  and  then  added  $2000  for  6 
months ;  B  put  in  $16000  for  3  months,  and  then  withdraw- 
ing half  his  capital,  continued  the  remainder  5  months 
longer;  C  put  in  $13500  for  7  months.  How  should  the 
profit  be  divided  ? 


EATIO  AND  PROPORTION.  177 

11.  A  and  B  entered  into  partnership  for  3  years,  A  put- 
ting in  15000,  and  B  $6000.  At  the  end  of  a  year  A  put 
in  $3000,  and  B  put  in  flOOO.  At  the  end  of  the  second 
year  A  took  out  $4000.  At  the  end  of  the  third  year  they 
divided  a  profit  of  $8140.    What  was  each  partner's  share  ? 

12.  A  and  B  engaged  in  trade  for  1  year.  Jan.  1st  A 
advanced  $2400  and  B  $3600 ;  May  1st  C  was  admitted  to 
the  firm  with  $4000;  July  1st  B  withdrew  $1000;  and 
Oct.  1st  C  withdrew  $150l).  Their  profits  for  the  year 
were  $6800 ;  what  was  each  partner's  share  ? 

13.  A  and  B  began  business  Jan.  1st,  each  with  a  capital 
of  $2500.  Apr.  1st  A  added  $500,  and  Aug.  1st  he  added 
$800  more.  June  1st  B  added  $1000.  What  was  the 
share  of  each,  at  the  year's  end,  of  a  profit  of  $5425  ? 

14.  A's  gain  is  $840,  B's  gain  is  $1125,  and  C's  gain  is 
$1820.  A's  capital  was  in  trade  7  months,  B's  9  months, 
and  C's  14  months.  How  much  of  the  capital  $13875  did 
each  own  ? 

Averages  or  Alligation. 

111.  The  process  of  finding  the  average  or  mean  value  of 
several  quantities  of  different  values  is  called  alligation 
medial. 

I.  A  grocer  mixed  12  lb.  of  tea  worth  40  cents  a  pound, 
10  lb.  worth  65  cents  a  pound,  and  8  lb.  worth  75  cents  a 
pound ;  what  was  the  mixture  worth  a  pound  ? 

40  X  12  =  480  12  lb.  at  40  cts.  a  pound  are  worth  480  cts. ; 

65  X  10  =  650  10  lb.  at  65  cts.  a  pound  are  worth  650  cts. ;  8  lb. 

75  X    8  =  600  at  75  cts.  a  pound  are  worth  600  cts.    Adding, 

30  )1730  we  find  the  value  of  30  lb.  to  be  1730  cts. ;  hence 

67|cts.  1  lb.  is  worth  3L  of  1730  cts.,  or  57f  cts. 


178  ARITHMETIC. 


EXAMPLES. 

1.  Four  children  weigh  respectively  62  lb.,  77  lb.,  89  lb., 
and  102  lb. ;  find  their  average  weight. 

2.  In  a  certain  school  there  are  15  pupils  10  years  old, 
6  pupils  9  years  old,  10  pupils  8  years  old,  8  "pupils  7  years 
old,  and  3  pupils  6  years  old ;  find  their  average  age. 

3.  Find  the  average  daily  expenses  of  a  travelling  sales- 
man whose  expenses  for  the  week  were  as  follows  :  Monday 
f  10.50,  Tuesday  $3.84,  Wednesday  $5.25,  Thursday  $4.33, 
Friday  $6.78,  and  Saturday  $9.44. 

4.  A  grocer  mixed  16  lb.  of  coffee  worth  25  cents  a 
pound,  24  lb.  worth  30  cents  a  pound,  and  10  lb.  worth  33 
cents  a  pound ;  what  was  the  mixture  worth  a  pound  ? 

5.  Teas  are  mixed  as  follows :  40  lb.  worth  70  cents  a 
pound,  60  lb.  worth  60  cents  a  pound,  100  lb.  worth  50  cents 
a  pound,  and  80  lb.  worth  40  cents  a  pound ;  for  what 
should  the  mixture  be  sold  a  pound  ? 

6.  Find  the  value  per  gallon  of  the  following  mixture : 
6  gal.  of  wine  worth  $1.10  per  gallon,  14  gal.  of  wine  worth 
$1.35  per  gallon,  7  gal.  of  wine  worth  $1.50  per  gallon,  and 
5  gal.  of  water. 

7.  A  merchant  sold  75  bbl.  of  flour  at  $5.60  per  barrel, 
45  bbl.  at  $5.95  per  barrel,  30  bbl.  at  $6.10  per  barrel,  and 
25  bbl.  at  $6.50  per  barrel ;  what  was  the  average  price  per 
barrel  ? 

8.  A  goldsmith  combined  7  oz.  of  gold  22  carats  fine,  12 
oz.  20  carats  fine,  10  oz.  15  carats  fine,  and  5  oz.  of  alloy ; 
how  many  carats  fine  was  the  composition  ? 

9.  A  miller  mixes  18  bu.  of  wheat  at  $1.44  with  6  bu. 
at  $1.32,  6  bu.  at  $1.08,  and  12  bu.  at  $0.84.  What  will 
be  his  gain  per  bushel  if  he  sells  the  mixture  at  $1.50  ? 


RATIO   AND   PROPORTION.  179 

10.  Some  sugar  is  adulterated  as  follows :  y^^  is  worth  8 
cents  per  pound,  |  is  worth  10  cents  per  pound,  ^^  is  worth 
12  cents  per  pound,  and  the  remainder,  33  pounds,  is  sand. 
What  is  the  mixture  worth  per  pound  ?  , 

112.  The  process  of  finding  the  proportion  of  several 
quantities  that  may  be  used  to  form  a  mixture  of  given 
average  value  is  called  alligation  alternate. 

I.  rind  the  proportion  in  which  teas  worth  respectively 
65  and  80  cents  a  pound  must  be  taken  to  form  a  mixture 
worth  70  cents  a  pound. 


65  +    5  .  .  10  .  .  2  Write  the  given  prices  in  a  column 

80  —  10  .  .     5  .  .  1  with  the  price  of  the  mixture  at  the  left. 

If  tea  worth  65  cts.  be  sold  for  70  cts., 

there  is  a  gain  of  5  cts.,  which  is  indi- 


70 

Ans.  2  lb.  at  ^^  cts 


and  1  lb.  at  80  cts.  ^^^^^  \iy\^  annexed  to  65 ;  if  tea  worth 

80  cts.  be  sold  at  70  cts.,  there  is  a  loss  of  10  cts.,  which  is  indicated  by 
— 10  annexed  to  80.  A  gain  of  5  cts.  a  pound  on  10  lb.  will  exactly  bal- 
ance a  loss  of  10  cts.  a  pound  on  5  lb.,  and  we  write  10  opposite  65  +  5, 
and  5  opposite  80  —  10.  This  means  that  the  two  kinds  must  be  mixed 
in  the  ratio  of  10  to  5;  dividing  both  numbers  by  5,  we  obtain  2  lb.  of 
the  first  and  1  lb.  of  the  second.  We  can  take  any  number  of  pounds 
that  are  in  the  ratio  of  2  to  1. 

Notice  that  the  number  of  pounds  taken  at  first  of  either  kind  is  the 
same  as  the  number  of  cents  gained  or  lost  on  the  other  kind. 

II.  Find  the  proportion  in  which  two  kinds  of  vinegar 
worth  respectively  12  and  15  cents  a  quart  must  be  taken 
to  form  a  mixture  worth  13 1  cents  a  quart. 

Using  the  same  process  as  in  the  pre- 
12  + 1^  •  •  1|^  •  •  5        ceding  problem,  we  find  that   the   two 

-IK -12.        11        4        kinds  are  to  be  mixed  in  the  ratio  of  1| 

to  U.    This  ratio  can  be  simplified  by 
Ans.  5  qt.  at  12  cts.  multiplying  both  numbers  by  3,   thus 

and  4  qt.  at  15  cts.  obtaining  5  qt.  of  the  first  kind,  and  4  qt. 

of  the  second  kind. 


l^ 


180 


ARITHMETIC. 


32 


22+10 

30+2 
35-3 

40-   8 J 


Ans.  4  lb.  at  22  cts., 
3  lb.  at  30  cts., 
2  1b.  at  35  cts., 
5  lb.  at  40  cts. 


32 


22  + 10-1 
30+   2— 


35 

40 


3J 


3 

8 
10 

2 


III.  A  grocer  wishes  to  mix  coffees  worth  respectively  22, 
30,  35,  and  40  cents  a  pound  so  as  to  make  a  mixture  worth 
32  cents  a  pound ;  how  many  pounds  of  each  kind  shall  he 
take  ? 

We  begin  as  before,  and  link  them 
together  two  by  two,  always  linking  one 
on  which  there  is  a  gain  with  one  on 
which  there  is  a  loss.  Comparing  the 
first  and  fourth,  we  find  that  they  must 
be  taken  in  the  ratio  of  4  to  5 ;  compar- 
ing the  second  and  third,  we  find  that 
they  must  be  taken  in  the  ratio  of  3  to  2. 
We  can  take  any  quantities  provided 
that  the  first  and  fourth  are  in  the  ratio 
of  4  to  5,  and  the  second  and  third  in  the 
ratio  of  3  to  2. 

By  linking  the  first  with  the  third, 
and  the  second  with  the  fourth,  we  ob- 
tain an  entirely  different  answer. 

When  more  than  two  kinds  are  to  be 
mixed,  they  can  be  taken  in  an  infinite 
number  of  ways,  for  it  is  only  necessary 
to  combine  them  two  by  two,  so  as  to 
make  mixtures  of  the  required  value,  and  tliese  mixtures  may  be  com- 
bined in  any  proportions  whatever.  In  solving  problems  it  is  generally 
best  to  take  the  combinations  that  involve  the  smallest  numbers. 

IV.  How  much  sugar  worth  respectively  6  and  10  cents 
a  pound  must  be  mixed  with  20  lb.  worth  9  cents  a  pound 
in  order  that  the  mixture  may  be  worth  8  cents  a  pound  ? 

1  .  .  10,  2  .  .  1  ;  10  +  1  =  11.  Link  the  first  with 

2  .  .  20  the    second    and    the 
2  .  .  1                                    first   with    the    third. 

Ans.   11  lb.  at  6  cts.  and  1  lb.  at  10  cts.         Comparing    the    first 

and  second,  we  find 
that  they  must  be  taken  in  the  ratio  of  1  to  2 ;  hence  10  lb.  of  the  first 
must  be  taken  with  20  lb.  of  the  second.  Comparing  the  first  and  third, 
we  find  that  they  must  be  taken  in  the  ratio  of  1  to  1.  Thus  we  must 
take  10  lb.  of  the  first,  and  in  addition  equal  quantities  of  the  first  and 
third. 


Ans.  3  lb.  at  22  cts., 
4  lb.  at  30  cts., 
10  lb.  at  35  cts., 
1  lb.  at  40  cts. 


6  +  2^ 
9-1 
10-2 


RATIO   AND  PROPORTION^  181 

V.   A  trader  mixed  oats  worth  respectively  32,  35,  40, 

xnd  42  cents  a  bushel  in  order  to  make  a  mixture  of  45  bu. 

worth  36  cents  a  bushel ;  how  many  bushels  of  each  kind 

did  he  take  ? 

We  find  that  the  dif- 
32  +  4-6..  3      3  +  4+1  +  2=10;  f^^^„,    kinda    may    be 


36 


35  +  1 
40-4 
42 -6J 


^                3          ^      27  mixed  in  quantities  pro- 

]        n      J^  ^^  ^^  "  2" "  ^^^  '  portional  to  3,  4,  1,  and 

^  •  •  ^       2  2.      Dividing    45    into 

2  parts    proportional     to 

JL    f  4S— IR"  these  numbers,  we  ob- 

;p  °    ^^~      '  tain  13.>  bu.  of  the  first 

?•  kind,  18  bu.  of  the  sec- 

9 

J_     f/er7_^_Ai.  ond,  4  J  bu.  of  the  third, 

X0^            2~    ^'  and  9 bu.  of  the  fourth. 

2  If    integral   answers 

j_    f  43  —  Q  *^®  desired,  the  ratios 

X0                   *  must  be  so  taken  that 

f  the  sum  of  the  numbers 

Ans.   13|bu.at32cts.,18bu.at35cts.,  representing  them  is  a 

4|bu.at40cts.,and9bu.at42cts.  ^^"^'°''   ^^    '^''   "°*^" 


quantity. 


EXAMPLES. 


1.  How  shall  corn  at  45  cts.  a  bushel  be  mixed  with  oats 
at  36  cts.  a  bushel  that  the  mixture  may  be  worth  40  cts.  a 
bushel  ? 

2.  Find  the  proportion  in  which  oils  worth  respectively 
$1.30  and  85  cts.  a  gallon  must  be  taken  to  form  a  mixture 
worth  $1.10  a  gallon. 

3.  In  what  proportion  shall  sugars  worth  respectively  7 
and  12  cents  a  pound  be  taken  to  form  a  mixture  worth  9^ 
cents  a  pound  ? 

4.  Find  the  proportion  in  which  sugars  worth  respec- 
tively 5  and  8  cents  a  pound  must  be  taken  to  form  a 
mixture  worth  6f  cents  a  pound. 


182  ARITHMETIC. 

5.  In  what  proportion  must  alcohol  (sp.  gr.  0.82)  be 
mixed  with  water  to  make  a  mixture  having  a  specific 
gravity  of  0.9  ? 

6.  A  grocer  wishes  to  mix  teas  worth  respectively  40, 

55,  and  65  cents  a  pound  so  as  to  make  a  mixture  worth  50 
cents  a  pound;  how  many  pounds  of  each  kind  shall  he  take  ? 

7.  Find  the  proportion  in  which  three  kinds  of  rice 
worth  respectively  S^,  11,  and  12 1  cents  a  pound  must  be 
taken  to  form  a  mixture  worth  10|^  cents  a  pound. 

8.  A  merchant  has  coats  worth  $12  each,  vests  worth  $6 
each,  and  hats  worth  $4^-  each ;  how  many  of  each  must  he 
sell  in  order  that  the  average  price  may  be  ^7^? 

9.  How  much  water  must  be  mixed  with  wine  worth  90 
cts.  per  gallon  to  make  a  mixture  worth  60  cts.  per  gallon  ? 

10.  A  grocer  wishes  to  mix  syrups  worth  respectively  42, 

56,  64,  and  75  cents  a  gallon  so  as  to  make  a  mixture  worth 
60  cents  a  gallon ;  how  many  gallons  of  each  kind  shall  he 
take  ? 

11.  In  what  proportion  shall  sugars  worth  respectively  6, 
7^,  8 1",  and  9  cents  a  pound  be  taken  to  form  a  mixture  worth 
7  cents  a  pound  ? 

12.  A  grocer  has  wines  worth  respectively  f  1.10,  $1.30, 
$1.35,  and  $1.50  per  gallon,  which  he  wishes  to  mix  with 
water  so  as  to  form  a  mixture  worth  $1.25  per  gallon ;  how 
many  gallons  of  each  shall  he  take  ? 

13.  Teas  at  3  s.  6  d.,  4  s.,  and  6  s.  a  pound  are  mixed  to 
produce  a  tea  worth  5  s.  a  pound ;  what  is  the  least  integral 
number  of  pounds  that  the  mixture  can  contain  ? 

14.  How  ma'  y  acres  of  land  worth  $70  an  acre  must  be 
added  to  a  fai  q  of  75  acres  worth  $100  an  acre  in  order 
that  the  whole  may  average  $80  per  acre  ? 


RATIO   AND   PROPORTION.  183 

15.  How  many  bushels  of  corn  at  50  cents  a  bushel  must 
be  mixed  with  100  bu.  of  oats  at  80  cents  a  bushel  that  the 
mixture  may  be  worth  75  cents  a  bushel  ? 

16.  How  many  pounds  of  chicory  at  6  cts.  a  pound  and 
coffee  at  28  cts.  a  pound  must  be  mixed  with  30  lb.  of 
coffee  worth  35  cts.  a  pound  in  order  that  the  mixture  may 
be  worth  20  cts.  a  pound  ? 

17.  A  goldsmith  has  4  oz.  of  gold  20  carats  fine  and  6  oz. 
22  carats  fine ;  how  many  ounces  of  alloy  must  be  combined 
with  it  to  make  a  mixture  16  carats  fine  ? 

18.  A  farmer  wishes  to  mix  20  bu.  of  oats  worth  33  cents 
a  bushel  with  oats  worth  respectively  35,  38,  and  40  cents  a 
bushel,  making  a  mixture  worth  36  cents  a  bushel;  how 
many  bushels  of  each  kind  must  he  take  ? 

19.  A  grocer  wishes  to  mix  15  lb.  of  coffee  at  40  cents  a 
pound  and  25  lb.  at  35  cents  a  pound  with  two  kinds  worth 
respectively  25  and  28  cents  a  pound  so  that  the  mixture 
may  be  worth  30  cents  a  pound ;  how  much  of  the  latter 

kinds  must  he  take  ? 

• 

20.  A  wholesale  dealer  has  an  order  for  1000  bu.  of  wheat 
at  75  cents  a  bushel ;  how  shall  he  mix  three  kinds  of 
wheat,  valued  respectively  at  72,  76,  and  80  cents  a  bushel, 
to  fill  the  order  ? 

21.  How  many  pounds  of  tea  worth  respectively  50,  60, 
and  75  cents  a  pound  must  be  taken  to  make  a  mixture  of 
70  lb.  worth  65  cents  a  pound  ? 

22.  A  man  has  100  three-cent  pieces,  which  he  wishes  to 
exchange  for  dimes,  half-dimes,  two-cent  pieces,  and  cents, 
and  still  have  the  same  number  of  coins ;  how  many  of  each 
kind  will  he  receive  ? 


184  •  ARITHMETIC. 

23.  A  lady  bought  100  yd.  of  cloth  for  $10,  some  at  3 
cents  a  yard,  some  at  8  cents,  some  at  12  cents,  and  some  at 
15  cents  ;  how  many  yards  of  each  kind  did  she  buy  ? 

24.  A  dealer  paid  f  182  for  20  barrels  of  flour,  giving  $10 
for  first  quality  and  $7  for  second ;  how  many  barrels  were 
there  of  each  ? 

25.  An  alloy,  formed  from  two  metals  whose  specific 
gravities  are  8.29  and  10.35,  has  a  specific  gravity  of  9.87  ; 
how  many  grams  of  each  metal  are  there  in  a  kilogram  of 
the  alloy  ? 

26.  Find  how  much  gold  15,  17,  and  22  carats  fine  must 
be  mixed  with  5  oz.  18  carats  fine  so  as  to  make  12  oz.  20 
carats  fine. 

27.  A  man  paid  $70  to  three  men  for  35  days'  labor ;  to 
the  first  he  paid  $5  per  day,  to  the  second  $1  per  day,  and 
to  the  third  $0.50  per  day.    How  many  days  did  each  work  ? 


PERCENTAGE.  186 


CHAPTER  IX. 

PERCENTAGE. 

113.  The  process  of  computing  by  hundredths  is  called 
percentage.  Per  cent,  a  contraction  of  the  Latin  per  centum^ 
means  hy  the  hundred.    For  example,  4  per  cent  of  25  means 

4  hundredths  of  25. 

The  sign  %  is  used  in  place  of  the  words  per  cent.  For 
example,  6%  means  6  per  cent.  6%  has  the  same  value  as 
0.06  or  ^. 

The  number  on  which  the  percentage  is  reckoned  is  called 
the  base,  the  number  of  hundredths  taken  is  called  the  rate, 
the  fraction  denoting  the  number  of  hundredths  is  called 
the  rate  per  cent,  and  the  part  of  the  base  corresponding 
to  the  rate  per  cent  is  called  the  per  cent  or  percentage. 
For  example,  5%  of  200  =  0.05  of  200  =  10 ;  200  is  the  base, 

5  is  the  rate,  5%  or  0.05  is  the  rate  per  cent,  and  10  is  th9- 
per  cent  or  percentage. 

To  Express  a  Rate  Per  Cent  as  a  Common  Fraction. 

114.  I.   Express  31:^%  as  a  common  fraction. 

5  Since   any   per  cent  equals  the 

3U%=^=3^X  — =  — •         same    number  of    hundredths,  31i% 
^'°     100       4       XPP     16  311 

4  equals  — i,  which  equals  ^^. 

EXAMPLES. 

Express  as  common  fractions  the  following  rates  per  cent : 

1.  6i%.  3.   ^%.  5.   1H%.  7.   14f%. 

2.  6|%.  4.   10%.  6.   12^%.  8.   16|%. 


186  ARITHMETIC. 

19.  80%. 

20.  831%. 

21.  87i%. 

22.  6%. 

23.  71%. 

24.  28%. 

25.  35%. 

26.  68f%. 

27.  125%. 

28.  1331%. 

Note.  The  fractional  equivalents  of  the  first  twenty-one  examples 
given  above  should  be  committed  to  memory  by  the  student,  as  they 
are  very  often  made  use  of. 

To  Express  a  Common  Fraction  as  a  Rate  Per  Cent. 

115.   I.   Express  f  as  a  rate  per  cent. 

2  _  28*  —  28  4  <^  Since  we  are  to  obtain  the  rate  per  cent,  or 

7  /  •  number  of  hundredths,  to  which  f  is  equal,  we 
divide  the  numerator  by  the  denominator,  carrying  the  division  to  two 
decimal  places  and  retaining  the  remainder  as  a  fraction. 

EXAMPLES. 

Express  as  rates  per  cent  the  following  common  fractions : 


9. 

20%. 

10. 

25%. 

11. 

331%. 

12. 

371%. 

13. 

40%. 

14. 

50%. 

15. 

60%. 

16. 

621%. 

17. 

66io/„ 

18. 

75%. 

29. 

140%. 

30. 

162^%. 

31. 

266|%. 

32. 

340%. 

33. 

Wc- 

34. 

-h% 

35. 

%% 

36. 

*%. 

37. 

t\%. 

38. 

\%% 

1.  \. 

6.   A- 

11.  i^. 

16.  ^f^. 

2.   J. 

7.  |. 

12.   W 

17.  ^. 

3.  \. 

8.  A- 

13.  A- 

18.   f. 

4.   f. 

9-  «• 

14.    W 

19.  |. 

6.  f 

10.  tV- 

15.  T*Tr- 

20.  ->^. 

PERCENTAGE. 


187 


To  Find  any  Per  Cent  of  a  Numbeb. 
U6.   I.   What  is  62-1%  of  72  ? 


9 


62i%  =  |;|ofr?  =  45.    ^  „„,_  „„^,  ,,  ,5. 


Since  62|%  equals  f ,  62^%  of  72  equals 


II.   What  is  12%  of  f210? 

210 
0.12 


Since  12%  equals  0.12,  12%  of  $210  equnl.<  0.12  of  $210, 
which  is  $25.20. 

^25.20 

Note  1.  When  a  problem  can  be  done  mentally,  use  the  method  of 
example  I. ;  otherwise  use  the  method  of  example  II.  * 

Note  2.  When  a  number  is  said  to  be  a  certain  per  cent  more  or 
less  than  another  number,  the  second  number  should  be  taken  as  the 
base;  never  use  the  sum  of  two  numbers  thus  spoken  of  as  the  base. 


1.  What 

2.  What 
8.  What 

4.  What 

5.  What 


EXAMPLES. 
s8%  of  1200? 

s  76%  of  126.25  ? 

s7i%  of  $2450? 

slf%  of  1264? 

s45%  off? 

s75%  of  12i? 

s  33^%  of  16  gal.  2  qt.  ? 


0.  What 

7.  What 

8.  What 

9.  What 

10.  What 

11.  What 

12.  What 

13.'  How  many  pounds  are  there  in  f  %  of  6  cwt.  1  qr.  ? 

U  What  is  62^%  of  175%  of  20%  of  $576? 


s  371%  of  6  ft.  8  in.  ? 
s  100%  of  1760  bu.  ? 
s  120%  of  250  mi.  ? 
s  11%  of  1280  men  ? 
sf%  of  140  books? 


1^8  ARITHMETIC. 

15.  A  man  bought  a  house  for  $6750,  paying  25%  cash; 
how  much  remained  due  ? 

16.  A  farmer  had  320  sheep,  and  lost  6^%  of  them;  how 
many  had  he  left  ? 

17.  A  man  has  a  yearly  income  of  $1400,  and  pays  17-f  % 
of  it  for  house  rent ;  what  rent  does  he  pay  ? 

18.  If  an  ore  yields  62%  of  pure  iron,  how  many  pounds 
of  iron  can  be  obtained  from  a  ton  of  ore  ? 

19.  A  miller  charges  6%  for  grinding;  how  many  quarts 
will  he  take  when  he  grinds  25  bu.  ? 

20.  A  merchant  failed  and  paid  60%  of  his  debts ; 
how  much  was  received  by  a  creditor  to  whom  he  owed 
$2180  ? 

21.  If  a  yard  of  cloth  shrinks  41%  in  length  in  spong- 
ing, what  fraction  of  a  yard  will  it  measure  after  sponging  ? 

22.  The  amount  of  sunshine  recorded  in  a  certain  city 
in  the  month  of  April  was  33%  of  the  possible  amount, 
and  the  average  length  of  the  nights  in  that  month  is  10  hr. 
30  min. ;  find  the  number  of  hours  of  sunshine  in  the 
month. 

23.  A  man  has  a  capital  of  $12500 ;  he  puts  15%  of  it 
in  stocks,  33|^%  in  land,  and  25<fo  in  mortgages ;  how  many 
dollars  has  he  left  ? 

24.  A  man  had  $3250  in  the, bank;  he  drew  out  12%  of 
it  at  one  time,  and  later  15%  of  the  remainder;  how  much 
was  left  in  the  bank  ? 

25.  A's  salary  is  $1350,  and  B's  salary  is  11|%  more 
than  A's ;  what  is  B's  salary  ? 

26.  A  boy's  age  is  60%  less  than  his  father's  age,  and 
his  father  is  45  years  old ;  what  is  the  boy's  age  ? 


PERCENTAGE.  189 


To   Find   the  Base  when  any  Per   Cent  op  it  is 

KifOWN. 

117.  I.   54  is  12^%  of  what  number? 

54 

r^  Since  12 1%  equals  },  54  is  \  of  the  required  number;  hence 

'        ■     the  number  equals  8  times  54,  or  432. 

II.  55  is  83^%  of  what  number? 

e     11     g  Since  83  J  %  equals  f,  66  is  f  of  the  required 

^^-i--=  ^pX-  =  QQ.       number,  and  the  number  is  as  much  as  f  is 

contained  in  55,  which  equals  66. 
Note.   The  methods  of  examples  r.  and  II.  are  to  be  used  only  when 
the  rate  per  cent  equals  a  small  common  fraction, 

III.  455  is  52%  of  what  number  ? 

.52)455.00(875 
416 
ogQ  Since  52%  equals  0.52,  466  is  0.62  of  the  re- 

o^^  quired  number,  and  the  number  is  as  much  as 

'  nnfi  0.52  is  contained  in  455,  which  equals  876. 

260 

IV.  What  number  increased  by  28%  of  itself  equals  320  ? 

1  28^^320  00C250  '^  number  increased  by  28%  of  itself  equals 

oKa  128%  of  itself,  which  is  the  same  as  1.28  times 

— ^jr  the  required  number.     If  320  is  1.28  times  some 

number,  the  number  is  as  much  as  1.28  is  con- 

^  tained  in  320,  which  equals  250. 

V.  Having  sold  36%  of  my  land,  I  have  224  acres  left; 
how  much  had  I  at  first  ? 

.64)224.00(350  A. 

192  Since  36%  was  sold,  there  remained  the  differ- 
"■^20  ^"^^  between  100%  and  36%,  or  64%.    Using  the 
220  same  method  as  in  the  preceding  examples,  we 
T  find  that  224  A.  is  64%  of  350  A. 


190  AHITHMETIC. 

EXAMPLES. 

1.  204  is  24%  of  what  number  ? 

2.  9.24  is  11%  of  wliat  number  ? 

3.  7465  is  331%  of  what  number  ? 

4.  126  is  175%  of  what  number  ? 

5.  5Jg-  is  371%  of  what  number  ? 

6.  15  is  1%  of  what  number  ? 

7.  $5674.83  is  105%  of  what  sum  ? 

8.  0.024  T.  is  21%  of  how  many  pounds  ? 

9.  261  is  16%  more  than  what  number  ? 

10.  What  is  the  number  to  which  if  2%  of  itself  be  added 
the  sum  is  516  ? 

11.  What  fraction  increased  by  16|%  of  itself  equals  i^? 

12.  96  is  14f  %  less  than  what  number  ? 

13.  What  number  diminished  by  36  %  of  itself  equals  336  ? 

14.  What  fraction  diminished  by  28%  of  itself  equals  i  ? 

15.  A  clerk  spends  $672  a  year,  which  is  56%  of  his 
salary ;  what  is  his  salary  ? 

16.  A  bankrupt  has  $5760,  and  with  that  sum  can  pay 
40%  of  his  debts ;  find  the  entire  amount  of  his  indebtedness. 

17.  The  average  daily  attendance  in  a  certain  school  is 
95,  which  is  83^%  of  the  entire  number;  how  many  pupils 
are  there  in  the  school  ? 

18.  For  collecting  a  bill  an  attorney  received  $2.52,  which 
was  l|-%  of  the  bill ;  what  was  the  amount  of  the  bill? 

19.  A  man  drew  out  37^%  of  his  bank  deposit  to  pay  a 
bill  of  $675 ;  how  much  had  he  remaining  in  the  bank  ? 


PERCENTAGE.  191 

20.  A  milkman  sold  milk  at  7  cents  a  quart,  which  was 
233-^%  of  the  cost ;  what  did  it  cost  a  gallon  ? 

21:  The  population  of  a  certain  town  has  gained  25% 
within  the  last  five  years.  It  is  now  6575 ;  what  was  it  five 
years  ago  ? 

22.  A  man  who  weighs  80*^^  is  220%  heavier  than  his 
son ;  what  is  the  weight  of  the  son  ? 

23.  A  regiment  lost  6J%  of  its  men  and  had  675  remain- 
ing ;  how  many  were  there  at  first  ? 

24.  Having  sold  30%  of  my  land,  I  have  17  acres  remain- 
ing ;  how  much  had  I  at  first  ? 

25.  The  expenses  of  a  concert  were  37|^%  of  the  receipts, 
and  the  profits  amounted  to  f  375 ;  what  were  the  expenses  ? 

26.  A  man  owning  33^%  of  a  factory  sold  60%  of  his  share 
for  $9000 ;  what  was  the  value  of  the  entire  property  at  the 
same  rate  ? 

27.  A  merchant  bought  some  flour  and  sold  30%  of  it  to 
one  customer  and  33  J  %  of  the  remainder  to  another,  and 
then  had  1120  barrels  remaining ;  how  many  barrels  did  he 
buy? 

28.  The  ages  of  two  brothers  together  amount  to  42 
years,  and  the  age  of  the  younger  is  75  %  of  the  age  of  the 
older ;  what  is  the  age  of  each  ? 

29.  The  number  of  pupils  in  a  school  is  378,  and  the 
number  of  boys  is  25  %  more  than  the  number  of  girls ; 
how  many  boys  are  there  ?  how  many  girls  ? 

30.  Two  farmers  together  own  540°*  of  land,  and  one 
farm  is  20  %  smaller  than  the  other ;  what  is  the  size  of 
each  farm  ? 


192  ARITHMETIC. 

To  Find  what  Pee  Cent  one  Number  is  of  Another. 

118.  I.   What  per  cent  of  90  is  60  ? 

60  _  2  _nQ2o/  ^^  ^^  t§  of  90.    Reducing  f§  to  its  lowest  terms, 

90       3  ^  we  obtain  f,  which  equals  66f%. 

II.   What  per  cent  of  |378  is  |135  ? 

378)135.00(.35f  =  35^%. 

^^^^  $135  is  iff  of   $378.    To  reduce 

2160  135  to  a  rate  per  cent,  we  divide  the 

1890  numerator  by  the  denominator,  carry- 

270  _  30  _  5  ing  the  division  to  two  decimal  places. 

378~42~7 

EXAMPLES. 

1.  What  per  cent  of  6  is  3  ? 

2.  What  per  cent  of  5  is  1  ? 

3.  What  per  cent  of  16  is  6  ? 

4.  What  per  cent  of  25  is  18  ? 

5.  What  per  cent  of  92  is  23  ? 

6.  What  per  cent  of  100  is  22^? 

7.  What  per  cent  of  3  is  J  ? 

8.  What  per  cent  of  |  is  f  ? 

9.  What  per  cent  of  1  score  is  1  dozen  ? 

10.  What  per  cent  of  f  40  is  $45  ? 

11.  What  per  cent  of  30000  bu.  is  50  bu.  ? 

12.  What  per  cent  of  |25  is  10  cts.  ? 

13.  What  per  cent  of  30  ft.  is  25  in.  ? 

14.  What  per  cent  of  1  rd.  3  yd.  2  ft.  5  in.  is  7  ft.  ? 


PERCENTAGE,  193 

15.  What  per  cent  of  f  105  is  30%  of  f  250  ? 

16.  What  per  cent  of  $2412  is  25%  of  $2856? 

17.  From  a  hogshead  of  molasses  containing  72  gal.,  6 
gal.  leaked  out ;  what  per  cent  was  lost  ? 

18.  If  a  house  worth  $1500  rents  for  $175  a  year,  what 
per  cent  of  its  value  is  the  rent  ? 

19.  If  33J  tons  of  iron  are  obtained  from  165  tons  of 
ore,  what  per  cent  of  the  ore  is  iron  ? 

20.  A  gold  ring  is  18  carats  fine ;  what  per  cent  of  it  is 

pure  gold  ? 

21.  If  a  city  in  five  years  increases  in  population  from 
23525  to  29171,  what  is  the  gain  per  cent  ? 

22.  A  clerk  has  a  salary  of  $1600,  and  his  expenses  for 
a  year  amounted  to  $1252  ;  what  per  cent  of  his  salary  did 
he  save  ? 

23.  A  pedlar  bought  30  doz.  pencils ;  after  selling  160, 
what  per  cent  of  the  original  number  remained  ? 

24.  A  man  bought  a  horse  and  carriage  for  $525,  paying 
$275  for  the  horse ;  what  per  cent  of  the  value  of  the  horse 
was  the  value  of  the  carriage  ? 

25.  If  carpet,  which  should  be  one  yard  wide,  is  only  34^ 
inches  wide,  what  per  cent  should  be  deducted  from  the 
price  ? 

26.  If  a  merchant  uses  a  false  weight  of  15J  oz.  instead 
of  a  pound,  what  per  cent  does  he  gain  by  his  dishonesty  ? 

27.  Of  an  alloy  containing  21  parts  copper  and  4  parts 
nickel,  what  per  cent  is  copper  ?  what  per  cent  is  nickel  ? 

28.  In  a  mixture  of  copper  and  zinc,  the  copper  is  to  the 
zinc  as  3^  is  to  2^ ;  express  the  percentage  of  each  ingredi- 
ent in  the  mixture. 


I 
194  ARITHMETIC. 

Profit  and  Loss. 

119.  Problems  in  percentage  dealing  with  gains  and 
losses  in  business  transactions  are  grouped  together  under 
the  subject  of  Profit  and  Loss.  The  cost  price  is  always  the 
base  in  determining  the  gain  or  loss  per  cent. 

EXAMPLES. 

1.  A  piece  of  cloth  containing  40  yards  cost  $5 ;  for 
what  must  it  be  sold  a  yard  to  gain  12|^%  ? 

2.  A  dealer  bought  lamps  at  $6.50  a  dozen,  and  sold 
them  at  a  loss  of  4% ;  what  did  he  receive  apiece  for  them  ? 

3.  A  grocer  bought  a  barrel  of  sugar  for  $14.75,  and 
sold  it  at  a  profit  of  8%  ;  what  was  his  gain  ? 

4.  A  grocer  bought  500  bags  of  coffee,  each  bag  contain- 
ing 49|-  pounds,  at  12  cents  a  pound,  and  sold  it  at  a  profit 
of  16|%  ;  what  did  he  receive  for  the  entire  amount  ? 

5.  I  buy  one  fifth  of  an  acre  of  land  for  $2178.  For 
how  much  a  square  foot  must  I  sell  it  in  order  to  gain  20% 
of  the  cost  ? 

6.  What  was  the  cost  when  17^%  was  gained  by  sell- 
ing an  article  for  $253.80  ? 

7.  Pind  the  cost  of  a  carriage  if  16|%  was  lost  by  sell- 
ing it  for  $250. 

8.  If  a  boy  sells  a  book  for  $1.10,  and  thereby  loses 
12%,  what  did  the  book  cost  him  ? 

9.  If  I  sell  coffee  at  2  s.  3  d.  per  pound,  and  thereby  gain 
35%,  what  did  I  give  per  pound  ? 

10.  What  per  cent  is  gained  in  buying  oil  at  80  cents  a 
gallon,  and  selling  it  at  12  cents  a  pint  ? 


PERCENTAGE.  195 

11.  What  per  cent  would  be  lost  in  buying  a  field  for 
$115,  and  selling  it  for  $100  ? 

12.  A  grocer  buys  sugar  at  18  cents  a  kilo,  and  sells  it 
at  1  cent  per  50*^ ;  how  much  per  cent  does  he  gain  ? 

13.  A  and  B  barter ;  A  makes  of  10  cents  12^^  cents,  and 
B  makes  of  15  cents  19  cents  j  which  makes  the  greater  per 
cent,  and  how  much  ? 

14.  If  I  sell  22  things  for  what  36  cost  me,  what  per  cent 

is  gained  ? 

15.  What  per  cent  is  gained  by  buying  coal  by  the  long 
ton,  and  selling  it  by  the  short  ton  at  the  same  price  ? 

16.  Having  purchased  an  acre  of  land,  I  sell  from  it  a 
rectangular  lot,  121  yards  long  and  25  yards  wide,  for  what 
the  whole  acre  cost  me;  what  per  cent  do  I  gain  on  the 
land  thus  sold  ? 

17.  If  the  cost  is  three  fourths  of  the  selling  price,  what 
is  the  gain  per  cent  ? 

18.  If  a  merchant  sells  goods  for  three  fourths  of  their 
cost,  what  per  cent  does  he  lose  ? 

19.  If  25%  of  the  amount  received  for  an  article  is  gain, 
what  is  the  gain  per  cent  ? 

20.  A  horse  that  cost  6J%  of  $25000  was  sold  for  $1000 ; 
what  was  the  loss  per  cent  ? 

21.  I  gained  33^%  in  selling  a  horse,  and  with  the  pro- 
ceeds bought  another  horse  which  I  afterwards  sold  for 
$120,  thereby  losing  25%  ;  did  I  gain  or  lose  by  the  trans- 
actions ? 

22.  If  a  wagon  is  purchased  at  20%  less  than  $50,  and 
is  afterwards  sold  at  25%  more  than  it  cost,  at  what  price 
is  it  sold  ? 


196  ARITHMETIC. 

23.  Four  coats  were  sold  for  $15  apiece  ;  a  profit  of  25% 
was  made  on  two  of  them,  and  on  the  other  two  25%  was 
lost ;  what  was  the  total  gain  or  loss  ? 

24.  A  merchant  selling  goods  at  a  certain  price  loses  5%  ; 
but  if  he  had  sold  them  for  $20  more,,he  would  have  gained 
3%  ;  what  did  the  goods  cost  him  ? 

25.  A  man  purchased  three  horses  for  $500.  The  first 
horse  cost  37|^%  less  than  the  second,  and  the  third  horse 
cost  60%  less  than  the  first ;  what  was  the  price  of  each  ? 

26.  If  25  pounds  of  tea  at  60  cents  per  pound  be  mixed 
with  30  pounds  at  49  cents,  find  the  price  of  the  mixture 
per  pound  in  order  that  there  may  be  a  profit  of  16 J%. 

27.  A  merchant  sold  one  half  of  a  certain  lot  of  goods 
at  a  gain  of  18%,  one  third  at  a  loss  of  5%,  and  the  re- 
mainder at  half  cost ;  did  he  gain  or  lose,  and  what  per 
cent  ? 

28.  A  man  buys  150  pounds  of  sugar,  and,  after  selling 
100  pounds,  finds  he  has  been  selling  it  at  a  loss  of  5%  ;  at 
what  rate  per  cent  advance  on  the  cost  must  he  sell  the 
remaining  50  pounds  that  he  may  gain  10%  on  the  entire- 
transaction  ? 

29.  A  speculator  had  5000  barrels  of  flour  that  cost  him 
$8  a  barrel;  he  sold  30%  of  the  lot  at  an  advance  of  10% 
on  the  cost,  and  50%  of  the  remainder  at  a  further  advance 
of  2^%  on  the  cost  j  he  closed  out  the  lot  at  $8.50  a  barrel. 
Find  how  much  he  made,  and  what  per  cent  of  the  cost. 

30.  A  merchant  sold  a  lot  of  flour  at  $8.40  a  barrel,  and 
thereby  gained  20%.  He  afterwards  sold  another  lot  of 
the  same  flour  for  $203,  and  thereby  gained  16%.  How 
many  barrels  were  there  in  the  last  lot  ? 


PERCENTAGE.  197 

120.   In  certain  problems  the  principles  of  proportion  can 
be  used  to  good  advantage. 

I.   If  I  sell  a  horse  for  $175  and  gain  5%,  what  would 
have  been  my  gain  per  cent  if  I  had  sold  it  for  $200  ? 

175  :  200  :  :  105  :  x.  A  gain  of  5%  is  the  same  as  105%  of  the 

g        25  cost.    Let  X  represent  the   per  cent  when 

_  %^^  X  X^^  _   OA  sold  for  .$200.     By  proportion  the  first  price 

~       %^^      ~       '  is  to  the  second  price  as  the  first  per  cent 

T  is  to  the  second  per  cent.     We  thus  obtain 

Arts.  20  %.  120%,  which  is  the  same  as  a  gain  of  20%. 

EXAMPLES. 

1.  If  there  is  a  gain  of  12^%  on  tea  at  90  cents  a  pound, 
what  would  be  the  gain  per  cent  at  84  cents  a  pound  ? 

2.  A  farmer  sold  a  pair  of  oxen  for  f  200,  and  gained 
20%  ;  what  per  cent  would  he  have  gained  if  he  had  sold 
them  for  $225  ? 

3.  By  selling  a  horse  for  $162,  a  man  lost  10%  ;  for 
how  much  should  he  have  sold  it  to  gain  10%  ? 

4.  If  5%  be  lost  by  selling  an  article  for  $2.47,  find  the 
per  cent  gain  or  loss  by  selling  it  at  $2.99. 

5.  A  man  lost  13|%  by  selling  a  lot  of  land  for  $850 ; 
what  would  it  have  brought  if  it  had  been  sold  at  a  loss  of 

6.  A  man  sells  flour  at  $6.50  a  barrel,  and  gains  10%  ; 
what  per  cent  would  he  gain  if  he  sold  the  flour  for  $8.25  a 
barrel  ? 

7.  By  selling  potatoes  at  62^  cents  a  bushel,  10%  was 
lost ;  at  how  much  should  they  be  sold  to  gain  25%  ? 

8.  A  man  sold  a  house  for  $5000,  and  thereby  gained 
20% ;  would  he  have  gained  or  lost,  and  how  much  per  cent, 
if  he  had  sold  it  for  $4000  ? 


198  ARITHMETIC. 

9.   If  I  lose  10%  by  selling  goods  at  28  cents  a  yard,  for 
what  should  they  be  sold  to  gain  20%  ? 

10.  A  village  lot  was  sold  for  $230,  which  was  8%  less 
than  it  cost ;  had  it  been  sold  for  $300,  what  would  have 
been  the  gain  per  cent  ? 

11.  A  firm  sold  a  fire  engine  for  $7050,  and  lost  6%  on 
its  cost ;  for  how  much  ought  they  to  have  sold  it  to  gain 
12%? 

12.  If  a  house  had  been  sold  for  $7992,  there  would  have 
been  a  gain  of  8  %  ;  how  much  per  cent  is  gained  or  lost  by 
selling  it  for  $7511  ? 

13.  If  50%  is  gained  by  selling  goods  at  3  s.  9d.  per 
pound,  what  per  cent  would  be  gained  or  lost  by  selling 
them  at  2s.  lid.  per  pound ? 

Commercial  Discount. 

121.  When  articles  are  sold  for  less  than  the  regular 
price,  the  difference  between  the  regular  price  and  the  price 
at  which  they  are  sold  is  called  the  discount,  and  the  price 
at  which  they  are  sold  is  called  the  net  price.  The  discount 
is  generally  a  certain  per  cent  of  the  regular  price. 

Manufacturers  and  wholesale  dealers  issue  price  lists 
subject  to  various  discounts. 

I.  What  is  the  net  price  of  a  carriage  billed  at  $275,  8% 
off  for  cash  ? 


275  275 

.08  22 


The  discount  is  8%  of  |275,  which  is  $22 ;  sub 
trading  this  from  $275,  we  find  the  net  price  to 
22.00  $253       ^e  $253, 

II.   Find  the  net  amount  of  a  bill  for  $762  subject  to  the 
following  discounts ;  40,  5,  and  10. 


PERCENTAGE.  199 

762  Allowing  a  discount  of  40%  is  the  same  as  tak- 

.60  ing  60%  of  the  amount  of  the  bill ;  hence  60%  of 

20H57  20  $762,  or  !$457.20,  is  the  amount  after  the  first  dis- 

oo  ofi  count.     Dividing  by  20  gives  6%  of  $457.20,  and 

'  subtracting  this  result,  we  find  $434.34  to  be  the 

-^  amount  after  the  second  discount.     Dividing  by 

43.43  10  gives   10%  of  $434.34,   and   subtracting  thig 

$390.91  result,  we  find  the  net  amount  to  be  $390.91. 

III.  At  what  per  cent  above  cost  must  a  merchant  mark 
his  goods  in  order  to  allow  a  discount  of  25%,  and  still 
make  a  profit  of  15%  ? 


If  he  makes  a  profit  of  16%,  he  sells  his  goods 
for  115%  of  the  cost.  If  he  allows  a  discount  of 
25%  from  the  marked  price,  115%  of  the  cost  is 
75%  of  the  marked  price.  Then  the  marked  price 
is  as  much  as  0.75  is  contained  in  115%,  which 
equals  153|%;  that  is,  the  goods  must  be  marked 
63^%  above  cost. 

EXAMPLES. 

1.  What  is  the  net  price  of  a  set  of  books,  the  list  price 
of  which  is  $24,  subject  to  a  discount  of  20%  ? 

2.  What  is  the  net  price  of  a  set  of  furniture,  the  list 
price  of  which  is  $225,  subject  to  a  discount  of  8J%  ? 

3.  When  a  piano,  whose  list  price  is  $850,  is  sold  at 
45%  off,  what  is  the  discount? 

4.  What  is  the  discount  on  a  bill  of  groceries  amounting 

to  $48.92,  7%  off  for  cash  ? 

5.  After  a  discount  of  20%  had  been  given,  a  refrigera- 
tor was  sold  for  $14.40 ;  what  was  the  list  price  ? 

6.  After  a  discount  of  6|%  had  been  given,  a  barrel  of 
flour  was  sold  for  $7 ;  what  was  the  list  price  ? 


.75)1.15(1.53^ 
75 
400 
375 

250 
225 

25 

75 

1 
'3 

Ans. 

53i%. 

200  ARITHMETIC. 

7.  What  is  the  net  price  of  a  bill  of  goods  amounting  to 
$170,  35%  discount  and  2^%  off  for  cash  ? 

8.  What  is  the  net  price  of  a  barrel  of  oil,  the  list  price 
of  which  is  $18,  subject  to  a  discount  of  12^%  and  4%  off 
for  cash  ? 

9.  Find  the  difference  between  the  net  value  of  a  bill  of 
goods  for  $1200,  less  a  discount  of  20%  and  5%  off  for 
cash,  and  the  net  value  of  the  same  bill,  less  a  discount  of 
25%. 

10.  Find  the  net  amount  of  a  bill  for  $234.60  subject  to 
the  following  discounts  :  50,  10,  and  7-^. 

11.  Find  the  net  amount  of  a  bill  for  $1036  subject  to 
the  following  discounts  :  30,  12^,  and  5. 

12.  Find  the  net  amount  of  a  bill  for  $856.25  subject  to 
the  following  discounts  :  40,  5,  15,  and  2^. 

13.  A  book  must  be  marked  at  what  per  cent  on  the 
original  cost  in  order  to  allow  a  discount  of  20%  of  the 
market  price,  and  still  give  10%  profit  ? 

14.  At  what  per  cent  above  cost  must  a  merchant  mark 
his  goods  in  order  to  allow  a  discount  of  33J%,  and  still 
make  a  profit  of  25%  ? 

15.  At  what  per  cent  above  cost  must  a  merchant  mark 
his  goods  in  order  to  allow  a  discount  of  16|%,  and  still 
make  a  profit  of  16f  %  ? 

16.  A  merchant  on  opening  a  case  of  goods  found  them 
slightly  damaged;  at  what  per  cent  above  cost  must  he 
mark  them  in  order  to  allow  a  discount  of  20%,  and  yet 
lose  only  8%  ? 

17.  A  house  was  bought  for  $500 ;  what  must  be  the 
asking  price  in  order  to  fall  on  it  10%,  and  still  make  10% 
on  the  purchase  ? 


PERCENTAGE.  201 

18.  A  merchant  bought  cloth  at  $3.60  a  yard ;  at  what 
price  must  it  be  marked  that  12^%  may  be  abated  from  the 
asking  price,  and  still  a  profit  made  of  16|%  ? 

19.  A  merchant  bought  carpeting  at  60  cents  a  yard.  At 
what  price  must  it  be  marked  in  order  that  in  selling  it  he 
may  abate  20%  on  the  marked  price,  and  yet  makd  a  profit 
of  33i%  ? 

20.  A  tradesman  marks  his  goods  at  25%  above  cost,- 
and  deducts  12%  of  the  amount  of  any  customer's  bill  for 
cash ;  what  per  cent  does  he  make  ? 

21.  A  merchant  marked  some  goods  30%  above  cost,  and 
then  sold  them  at  a  discount  of  25%  ;  did  he  gain  or  lose, 
and  what  per  cent  ? 

22.  If  from  the  retail  price  of  a  book  20%  is  deducted, 
and  a  discount  of  10%  is  made  on  the  balance,  and  then  the 
book  sells  for  $1.33,  what  is  the  retail  price  ? 

23.  A  manufacturer,  who  allows  a  discount  of  20%  from 
the  list  price  with  5%  off  for  cash,  receives  $262.20  as  the 
net  price  of  a  bill  of  goods  ;  find  the  list  price. 

24.  A  merchant  sold  a  quantity  of  goods  for  $29900. 
He  deducts  5%  from  the  amount  of  the  bill  for  cash,  and 
finds  that  he  has  made  15%  on  the  investment;  what  did 
he  pay  for  the  goods  ? 

Commission. 

122.  A  person  who  buys  or  sells  goods  for  another  is 
known  as  a  commission  merchant  or  agent.  The  compensa- 
tion received  is  called  commission,  and  it  is  usually  a  certain 
per  cent  of  the  amount  expended  or  collected. 

After  the  commission  and  other  charges  have  been  de- 
ducted from  the  amount  of  a  business  transaction,  the 
remainder  is  called  the  net  proceeds. 


202 


ARITHMETIC. 


I.  Find  the  net  proceeds  on  the  sale  of  500  bbl.  of  flour  at 
$4.25  per  barrel;  the  commission  for  selling  was  2J%,  and 
the  charges  for  freight  and  storage  were  32  cts.  per  barrel. 


4.25 
500 
2125.00 
.021 
53125 
425000 
47.8125 
160. 
207.81 


0.32 

500 
160.00 


2125.00 

207.81 

$1917.19 


At  14.25  per  barrel,  the  price  of  500 
bbl.  of  flour  is  $2125.  21%  of  $2125 
is  $47.81,  the  amount  of  the  commis- 
sion ;  at  32  cts.  per  barrel  the  charges 
for  freight  and  storage  amount  to  $160. 
The  sum  of  these  expenses  is  $207.81, 
which  subtracted  from  $2125  gives 
$1917.19  as  the  net  proceeds. 


II.  $1000  includes  a  sum  to  be  invested  and  a  commis- 
sion of  5%  of  the  sum  to  be  invested;  what  is  the  sum  to 
be  invested  ? 


1.05)1000.00(1952.38 
945 

550 

525 


250 
210 


400 

315 
850 
840 


Since  $1000  includes  both  the  sum  to 
be  invested  and  5%  of  that  sum,  it  is 
105%  of  the  investment.  If  $1000  is 
105%  of  the  investment,  the  investment 
is  as  much  as  1.05  is  contained  in  $1000, 
which  equals  $952.38. 


EXAMPLES. 


1.  At  an  auction  sale  $542.68  was  received  for  a  lot  of 
furniture ;  what  was  the  auctioneer's  commission  at  5%  ? 

2.  Find  the  commission  at  3|^%  on  the  sale  of  952  yards 
of  carpeting  at  62  cents  a  yard. 

3.  An  architect  charged  2^%  for  the  plans  and  specifi- 
cations of  a  building  costing  $60000,  and  2^%  for  superin- 
tending the  work  of  erection  ;  what  was  his  fee  ? 


PEKCKNTAGE.  203 

4.  A  real  estate  agent  sold  a  house  for  $7500,  and 
charged  f  %  commission  ;  find  the  net  proceeds  of  the  sale. 

5.  A  commission  merchant  sold  a  consignment  of  sugar 
for  $3420,  and  charged  2f  %  commission ;  the  expenses  for 
freight  and  storage  were  $24.70  j  find  the  net  proceeds. 

6.  A  commission  merchant  sold  1350  baskets  of  peaches 
at  $1.20  a  basket;  the  commission  for  selling  was  3f%, 
the  charge  for  guaranteeing  payment  was  1^%,  and  the 
charge  for  cartage  was  4  cents  a  basket ;  find  the  net  pro- 
ceeds. 

7.  An  agent's  commission  at  5%  amounted  to  $362.25; 
find  the  amount  of  his  sales. 

8.  If  I  sell  goods  at  4%  commission,  and  receive  $60, 
what  amount  have  I  sold  ? 

9.  A  travelling  agent,  who  received  5%  commission,  re- 
mitted $893  as  the  net  proceeds  for  a  week's  sales ;  find 
the  amount  of  his  sales. 

10.  A  real  estate  agent  received  8J%  commission  for 
selling  some  land,  and  forwarded  $4400  to  the  owner  as  net 
proceeds ;  for  how  much  was  the  land  sold  ? 

11.  An  attorney  received  $58.50  as  his  commission  for 
collecting  a  bill  of  $975 ;  what  was  the  rate  of  commission  ? 

12.  A  book  agent  sold  84  books  at  $3.50  apiece,  and  after 
deducting  his  commission,  remitted  $176.40  to  the  publish- 
ers ;  what  was  the  rate  of  his  commission  ? 

13.  An  agent  invested  $15000  in  cotton  for  a  manufac- 
turing company,  charging  a  commission  of  If  %  ;  what  was 
the  entire  bill  for  cotton  and  commission  ? 

14.  A  collector,  who  charges  8%  commission  on  what  lie 
collects,  pays  me  $534.75  for  a  bill  of  $775 ;  what  fractional 
part  of  the  bill  does  he  collect  ? 


204  AEITHMETIC. 

15.  An  agent  received  a  sum  of  money  to  expend,  aftei 
deducting  his  commission  of  3^%  ;  he  expended  $523.67  ; 
what  was  the  sum  he  received  ? 

16.  $1200  includes  a  sum  to  be  invested  and  a  commis- 
sion of  4%  of  the  sum  to  be  invested ;  what  is  the  sum  to 
be  invested  ? 

17.  A  commission  merchant  receives  $625  to  invest  in 
goods,  after  deducting  his  commission  of  4^%  ;  find  the 
amount  of  his  commission. 

18.  At  $6.25  per  ton,  how  many  tons  of  coal  can  I  buy 
for  $1000  and  allow  2-^%  commission  ? 

19.  An  agent  receives  $6150  to  invest  in  cotton  at  10^ 
cents  per  pound ;  his  commission  is  2-|-%  ;  how  many  pounds 
of  cotton  can  he  buy  ? 

20.  A  commission  merchant  received  $1356.60  to  be 
expended  in  flour  at  $4.75  a  barrel,  after  deducting  his  com- 
mission of  2%  ;  how  many  barrels  did  he  buy  ? 

21.  A  sends  B  $5000,  with  which  B  is  to  purchase  lum- 
ber, after  providing  for  his  commission  of  2%  ;  how  much 
will  B  have  to  expend  for  lumber,  and  what  will  be  the 
amount  of  his  commission  ? 

22.  An  agent  received  $1250,  with  which  to  purchase 
goods,  after  deducting  his  commission  of  2|-%,  and  paying 
$11.50  for  cartage ;  what  sum  did  he  pay  for  the  goods  ? 

23.  A  commission  merchant  sold  goods  for  $7125,  and 
received  4  %  commission ;  he  invested  the  net  proceeds  in 
lumber,  charging  2  %  commission  for  buying ;  find  the 
value  of  tlie  lumber  bought. 

24.  A  commission  merchant  sold  1050  pounds  of  butter 
at  24  cents  a  pound,  and  received  a  commission  of  2|  %  ; 
he  invested  the  net  proceeds  in  corn  at  45  cents  a  bushel, 
charging  1^  %  commission  j  how  many  bushels  of  corn  did 
he  buy  ? 


PERCENTAGE.  205 

25.  A  real  estate  agent  sold  a  lot  of  land  for  $8000,  and 
with  the  proceeds  bought  a  house,  after  deducting  his  com- 
mission of  21  %  for  selling  and  4  %  for  buying ;  what  was 
his  entire  commission? 


Insurance. 

123.  Insurance  is  a  contract  whereby  one  party  agrees 
for  a  specified  consideration  to  pay  a  specified  sum  of  money 
in  case  of  loss  by  certain  risks. 

In  fire  insurance  the  agreement  is  to  pay  for  all  losses  by 
tire  not  exceeding  a  certain  sum,  and  in  marine  insurance 
the  same  is  true  for  all  losses  arising  from  accidents  of 
navigation.  Life  insurance  secures  a  certain  sum  of  money 
to  a  person's  heirs  in  case  of  death. 

The  written  contract  is  called  the  policy,  and  the  sum 
paid  for  insurance  is  called  the  premium ;  the  premium  is 
computed  at  a  certain  per  cent  on  the  amount  insured,  or  at 
so  much  on  the  $100. 

I.  A  store  worth  $30000  was  insured  for  J  of  its  value 
at  IJ  %  ;  find  the  annual  premium. 

6)30000 

5000 

5 

25000  5  of  $30000  is  $25000;  1^  %  of  $25000  equals  $312.50. 


6250 

25000 

$312.50 

II.  For  how  much  must  a  cargo  of  lumber  worth  $18750 
be  insured  at  3^  %,  so  that  in  case  of  loss  the  owner  may 
recover  both  the  value  of  the  lumber  and  the  premium  ? 


206  '  ARITHMETIC. 

.965)18750.000(^19430.05 
965 

9100 

gggg  Since  the  premium  is  3^  %  of  the 

amount  insured,  the  value  of  the  lum- 
ber is  100% -31%,  or  961%,  ^f  ^he 
amount  insured.  Hence  the  entire 
amount  is  as  much  as  .965  is  contained 
in  $18750,  which  equals  $19430.05. 


4150 
3860 
2900 
2895 


5000 
4825 


EXAMPLES. 

1.  What  premium  must  be  paid  for  an  insurance  of 
$4500  on  a  house  at  1J%  ? 

2.  What  is  the  expense  of  insuring  a  mill  worth  $12000 
for  i  01  its  value  at  4^%  ? 

3.  If  the  premium  for  insurance  at  |%  is  $16.50,  what 
is  the  amount  insured  ? 

4.  A  stable  was  insured  for  ^  of  its  value  at  2J%, 
and  the  premium  was  $55;  what  was  the  value  of  the 
stable  ? 

5.  If  a  premium  of  $60  is  paid  for  an  insurance  of 
$4800  on  a  house,  what  is  the  rate  of  insurance  ? 

6.  A  merchant  paid  $20  for  an  insurance  of  $2500  on  a 
stock  of  goods ;  find  the  rate  of  insurance. 

7.  A  cargo  worth  $8150  was  insured  at  3f  %  for  -^  of 
its  value ;  in  case  of  shipwreck  what  would  be  the  actual 
loss  of  the  owner  ? 

8.  A  house  was  insured  for  $5500  at  2J% ;  during  the 
fourth  year  of  the  insurance  the  house  was  burned;. find 
the  actual  loss  of  the  insurance  company,  makinp^  ro  aJlow- 
ance  for  intere^^^ 


PERCENTAGE.  207 

9.  A  house,  which  had  been  insured  for  f  2500  for  12 
years  at  l|-%,  was  wholly  destroyed  by  fire;  how  much  did 
the  amount  received  from  the  company  exceed  the  sum  of 
the  premiums  paid  ? 

10.  For  what  sum  ought  a  ship  worth  $15200  to  be  in- 
sured at  5%,  so  that  in  case  of  loss  tlie  owner  may  recover 
both  the  value  of  the  ship  and  the  premium  ? 

1 1.  A  merchant  had  a  cargo  of  sugar  worth  $3200  in- 
sured at  2J%,  so  as  to  cover  both  the  value  of  the  sugar 
and  the  premium ;  for  how  much  was  the  sugar  insured  ? 

12.  Find  the  premium  paid  when  goods  worth  $9500  are 
insured  at  3%  ^o  as  to  cover  both  the  value  of  the  goods 
and  the  premium. 

13.  A  ship  was  insured  for  $25600  to  cover  both  the 
value  of  the  ship  and  the  premium  of  6J%  ;  find  the  value 
of  the  ship. 

14.  A  mill  is  insured  for  $8000  in  one  company  and  for 
$5000  in  another  company;  a  fire  causes  a  loss  of  $6500; 
what  amount  should  each  company  pay  ? 

15.  A  building  worth  $12500  is  insured  for  |  of  its  value 
in  three  companies;  the  first  takes  \  of  the  risk  at  J%,  the 
second  takes  |  of  it  at  1%,  and  the  third  takes  the  remainder 
at  l|-%  ;  what  is  the  total  premium? 

16.  If  the  building  in  the  preceding  problem  should  be 
damaged  to  the  amount  of  $2025,  what  ought  each  company, 
to  pay  ? 

17.  A  man  30  years  old  had  his  life  insured  for  $5000  at 
$25.25  per  $1000 ;  what  was  the  annual  premium  ? 

18.  If  a  man  pays  a  yearly  premium  of  $74.20  fo^-  life 
insurance  at  $21.20  on  $1000.  what  is  the  sum  insured ) 


208  ARITHMETIC. 

19.  A  man  had  his  life  insured  for  flOOOO  at  $32.40  pel 
$1000 ;  should  he  die  after  paying  eight  yearly  premiums, 
how  much  more  would  his  heirs  receive  than  had  been  paid 
in  premiums? 

Taxes. 

124.  A  tax  is  a  sum  of  money  assessed  on  the  person  or 
property  of  an  individual  for  public  purposes. 

Taxes  are  apportioned  among  the  tax-payers  according  to 
the  estimated  value  of  their  property. 

In  some  states  each  male  citizen  pays  a  certain  sum  with- 
out regard  to  his  property ;  this  is  called  a  poll  tax. 

In  computing  taxes  the  amount  of  the  poll  taxes,  if  any, 
is  deducted  from  the  entire  sum  to  be  raised ;  the  remainder 
divided  by  the  value  of  the  taxable  property  gives  the  rate 
of  taxation,  which  may  be  expressed  as  a  per  cent,  or  as 
so  much  on  $1000.  For  example,  if  $3000  is  to  be  raised 
by  taxing  property  worth  $600000,  the  rate  is  -J-%,  or  $5  on 
$1000. 

I.   A  tax  of  $15377.42  is  to  be  raised  in  a  town,  the  tax- 
able property  of  which  is  valued  at  $1672316;  there  are  528 
polls,  each  taxed  $2 ;  find  the  rate  of  taxation. 
528         15377.42 

?  ^^^^  The  tax  on  528  polls  at  $2  each 

1056  14321.42  amounts   to   $1056.       Subtracting 

this   amount   from    $15377.42,   we 

1672316)  14321.420  (.008564       find  114321.42  to  be  the  amount  to 

13378528  ^e  raised  by  taxing  the  property, 

9428920  ^   $14321.42    is   to   be   raised   on 

8361580  $1672316,  the  amount  to  be  raised 

10673400  ®^  ^^  ^®  ^®  much  as  1672316  is  con- 

100S3896  tained  in  $14321.42,  which  equals 

—     '  $0.008564;    this    is    the   same   as 

b,:5yoU4U  ^g  gg^  ^^  ^^^^ 

_xns.  $8,564  on  $1000, 


PERCENTAGE.  209 

II.  Using  the  rate  of  taxation  found  in  the  preceding 
problem,  find  the  tax  paid  by  a  man  who  pays  a  poll  tax 
and  owns  property  valued  at  $7132. 

7132 
.008564 
28528 
42792  If  the  tax  on  $1  is  .10.008564,  on  $7132  the  tax  is 

35660  7132  times  ."$0.0085(54,  which  equals  .^61.08.     Add- 

57056  ing  the  poll  tax,  we  find  the  entirt  tax  to  be  $03.08. 

61.078448 
2 


$63.08 


EXAMPLES. 


1.  Find  the  rate  of  taxation  when  $8647.29  is  to  be 
raised  by  taxing  property  valued  at  $768648. 

2.  If  the  assessed  valuation  of  a  town  is  $2362724,  and 
the  town  has  637  polls,  paying  $1.50  each,  what  must  be 
the  rate  of  taxation  in  order  to  raise  $24516  ? 

3.  A  tax  of  $9426.88  is  to  be  assessed  in  a  town ;  the 
real  estate  is  valued  at  $442300,  and  the  personal  property 
at  $496720;  there  are  486  polls,  paying  $1.25  each;  what 
is  the  rate  of  taxation  ? 

4.  A  tax  of  $7137  is  to  be  assessed  in  a  town  having 
610  polls  and  a  valuation  of  $817326 ;  if  one  sixth  of  the 
tax  be  laid  on  the  polls,  what  will  be  the  amount  of  each 
poll  tax,  and  what  will  be  the  rate  of  taxation  ? 

5.  At  the  rate  of  $13.70  on  $1000,  find  the  tax  paid  by 
a  man  who  pays  a  poll  tax  of  $1.80,  and  owns  property  val- 
ued at  $72540. 

6.  Find  the  tax  to  be  paid  by  a  man  who  pays  a  poll 
tax  of  $1.40,  and  owns  property  valued  at  $18732,  when  the 
rate  of  taxation  is  $14.32  on  $1000. 


210  ARITHMETIC. 

7.  A  man  owns  property  valued  at  $25420,  of  which 
$2400  is  exempt  from  taxation ;  find  his  tax  at  the  rate  of 
$10.26  on  $1000. 

8.  A  tax  of  $9426  is  to  be  raised  in  a  town,  the  taxable 
property  of  which  is  valued  at  $921495;  find  the  tax  on 
property  valued  at  $2340. 

9.  Find  the  rate  of  taxation  when  a  man,  who  owns 
property  valued  at  $28000^  pays  a  tax  of  $370.16. 

10.  A  man  pays  a  tax  of  $799.07,  including  a  poll  tax  ol 
$2 ;  the  rate  of  taxation  is  $16.30  on  $1000 ;  find  the  value 
of  his  property. 

11.  Find  the  value  of  the  taxable  property  of  a  town 
when  a  tax  of  $11131.02  is  raised  at  the  rate  of  $11.25  on 
$1000. 

12.  A  collector  receives  8%  for  collecting  taxes,  and  pays 
into  the  treasury  $94625.64  after  deducting  his  commission ; 
how  much  did  he  collect  ? 

13.  The  taxes  assessed  in  a  town  amounted  to  $34271.60; 
the  collector  received  1^%  for  collecting,  and  7%  of  the 
taxes  could  not  be  collected ;  find  the  net  proceeds. 

14.  Find  the  entire  tax  that  must  be  assessed  in  order 
that  a  town  may  receive  $12134  after  the  collector  deducts 
his  commission  of  2^%. 

15.  The  net  proceeds  of  the  taxes  assessed  in  a  town, 
after  deducting  the  collector's  commission  of  2%,  amounted 
to  $50638.95,  and  5%  of  the  taxes  could  not  be  collected; 
find  the  amount  of  taxes  assessed. 

16.  In  a  certain  city  the  cost  of  public  schools  for  the 
next  school  year  is  estimated  at  $36848.  Find  what  amount 
of  school  tax  must  be  assessed,  the  cost  of  collecting  being 
2%  of  the  assessed  tax,  and  allowing  6%  of  the  assessed 
tax  to  be  not  collectible. 


PERCENTAGE.  211 


Duties. 


125.  Duties  are  taxes  levied  .by  the  government  on  im- 
ported goods. 

An  ad  valorem  duty  is  a  certain  per  cent  of  the  cost  of 
the  goods  in  the  country  from  which  they  were  imported. 

A  specific  duty  is  a  tax  levied  upon  goods  according  to 
the  quantity,  without  reference  to  the  value.  In  calculating 
specific  duties,  allowances  are  made  (1)  for  the  weight  of 
the  box,  cask,  or  bag  containing  the  goods,  called  tare ; 
(2)  for  the  loss  of  liquids  in  barrels  or  casks,  called  leak- 
age ;  (3)  for  the  loss  of  liquids  in  bottles,  called  breakage. 
The  weight  of  goods  before  any  allowances  have  been  made 
is  called  the  gross  weight,  and  the  weight  after  all  allow- 
ances have  been  made  is  called  the  net  weight. 

I.  rind  the  duty,  at  35%  ad  valorem,  on  linen  goods 
invoicedat  £128  7s.  6d. 

12)6.  d.  624.74 

20)7.5s.  iM 

£128:375  312370 

4  8665  187422 

' 91 «  f\nQC\         "^^^  value  of  the  goods  in  United 

641875  ^l^.bSyU     g^^^^g  ^^^^^^  .g  $624.74,  and  35% 

770250  ouoiQ^c     of  $624.74  is  $218.66. 

770250       Ans.  1218.66. 

1027000  . 

513500  * 


$624.7369375 

II.    Find  the  duty    at  4  cents  a  gallon,  on  150  hhd.  of 
molasses,  63  gal.  in  a  hogshead,  leakage  2%. 

150  hhd.  equals  9450  gal.     The 

63  9450  leakage   is    2%  of    this    quantity, 

150  189  which  equals  189  gal.     This  leaves 

9450  9261  9261  gal.  on  which  duty  must  be 

.02  .04  paid.    At  4  cts.  a  gallon,  the  duty 


189.00  1370.44  on  9261  gal.  is  9261  times  4  cts. 

which  equals  $370.44. 


212  ARITHMETIC. 

EXAMPLES. 

1.  Find  the  duty,  at  50%  ad  valorem,  on  1650  yd.  of  silk 
valued  at  $1.85  per  yard. 

2.  Find  the  duty,  at  50  cts.  per  gallon,  on  40  casks  of 
sherry  wine,  each  containing  30  gal. 

3.  What  is  the  duty,  at  2  J  cts.  per  pound,  on  3500  lb.  of 
rice,  tare  5%  ? 

4.  What  is  the  duty,  at  3  cts.  per  pound,  on  80  hhd.  of 
sugar,  each  weighing  5001b.,  tare  12i%  ? 

5.  A  merchant  imported  a  lot  of  china,  invoiced  at 
£422  10  s. ;  what  was  the  duty  at  55%  ad  valorem  ? 

6.  Find  the  duty,  at  $2.50  per  pound  and  25%  ad 
valorem,  on  500  boxes  of  cigars,  each  containing  50  cigars, 
invoiced  at  $4.50  per  box,  and  weighing  12^- lb.  per  1000. 

7.  A  jeweller  imported  from  Geneva  40  doz.  watches, 
invoiced  at  312.50  francs  per  dozen;  find  the  duty  at  25% 
ad  valorem. 

8.  A  merchant  imported  1800  yd.  of  Brussels  carpeting 
I  yd.  wide,  invoiced  at  4  s.  6d.  per  yard;  find  the  duty  at 
30  cts.  per  square  yard,  and  in  addition  30%  ad  valorem. 

9.  Find  the  cost  per  dozen  if  the  duty  on  30  doz.  clocks 
at  30%  ad  valorem  is  $135. 

10.  A  merchant  imported  a  lot  of  needles  valued  at  $600, 
and  paid  $210  duty ;  what  was  the  rate  of  duty  ? 

11.  A  man  paid  $51.25,  including  a  duty  of  25%,  for  a 
watch ;  how  much  was  the  duty  ? 

12.  Find  the  entire  cost  of  a  lot  of  glassware  on  which  a 
duty  of  $623.70  was  paid;  the  duty  was  45%  ad  valorem, 
and  damages  of  16%  were  allowed  for  breakage. 


PERCENTAGE.  213 

MISCELLANEOUS    EXAMPLES. 

1.  The  difference  between  12^^%  and  16J%  of  a  number 
equals  60 ;  what  is  the  number  ? 

2.  What  per  cent  of  a  pound  Troy  is  a  pound  Avoirdu- 
pois ? 

3.  A  horse  costs  f  as  much  as  a  carriage ;  what  per  cent 
of  the  price  of  the  horse  was  the  price  of  the  carriage  ? 

4.  A  gain  of  25%  was  made  by  selling  flour  at  f  6  per 
barrel ;  at  what  price  ought  it  to  be  sold  to  gain  33|  %  ? 

5.  If  10%  is  lost  by  selling  boards  at  $7.20  per  M,  what 
per  cent  would  be  gained  by  selling  them  for  90  cents  per  C  ? 

6.  When  eggs  were  sold  at  83f%  of  the  cost,  there  was 
a  loss  of  5  cents  a  dozen ;  how  much  would  have  been  lost 
by  selling  them  for  80%  of  the  cost  ? 

7.  If  the  cost  be  three  fifths  of  the  selling  price,  what  is 
the  gain  per  cent  ? 

8.  A  man  sold  160  acres  of  land  for  $4563.20,  which 
was  8%  less  than  it  cost.     What  was  the  cost  per  acre  ? 

9.  A  man  bought  a  yacht  for  $3500,  and  sold  it  at  a  loss 
of  20%  ;  the  buyer  sold  it  at  a  gain  of  25%  ;  what  did  the 
latter  receive  for  it  ? 

10.  A  grocer  bought  a  barrel  (31^  gal.)  of  vinegar  for 
$5.95,  and  5f  %  of  it  leaked  out ;  for  how  much  a  gallon 
must  the  remainder  be  sold  in  order  to  gain  20%  ? 


11.  A  jeweller  sold  two  watches  for  $60  each;  on  one  he 
gained  25%,  and  on  the  other  he  lost  25%  ;  did  he  gain  or 
lose,  and  how  much  ? 

12.  A  clock,  marked  40  %  above  cost,  was  sold  at  a  reduc- 
tion of  40%  ;  what  was  the  gain  or  loss  per  cent  ? 


214  ARITHMETIC. 

13.  A  man  sold  two  horses,  which  he  had  previously  pur- 
chased, for  $200  each.  On  one  of  them  he  made  a  profit  of 
33|^%,  and  on  the  other  he  incurred  a  loss  of  20%.  Did  he 
gain  or  lose  in  the  transaction  ?  What  did  each  horse  cost 
him  ? 

14.  A  horse  worth  f  250  was  bought  for  $25  less,  and 
sold  for  $25  more  than  its  real  value ;  what  was  the  gain 
per  cent  ? 

15.  If  a  piece  of  land  is  bought  for  $3500,  and  a  man 
who  owns  "I  of  it,  sells  one  half  of  his  share  for  $800,  how 
much  per  cent  does  he  gain  by  the  transaction  ? 

16.  75%  of  the  area  of  a  farm  is  arable;  of  the  remainder 
85%  is  pasture,  and  the  rest  is  waste;  if  the  area  of  the 
waste  is  3  A.  20  sq.  rd.,  what  is  the  area  of  the  farm  ? 

17.  For  each  of  three  successive  years  the  population  of 
a  town  rose  50%,  and  at  the  end  of  the  third  year  it  was 
2700;  what  was  the  population  at  the  beginning  of  the 
time? 

18.  The  population  of  a  city  in  1871  increased  4%  on 
that  of  1870,  in  1872  it  increased  5%  on  that  of  1871,  and 
in  1873  it  increased  6%  on  that  of  1872  and  amounted  to 
1389024 ;  find  the  population  in  1870. 

19.  A  man  invests  a  sum  of  money  in  such  a  way  that  it 
gains  10% ;  he  adds  the  gain  to  the  original  sum,  and  invests 
the  whole  in  such  a  way  as  to  gain  15%  ;  the  entire  amount 
is  then  $50600 ;  what  is  the  amount  originally  invested  ? 

20.  A  man  put  $780  in  the  bank,  which  was  15%  of  all 
his  money ;  he  afterwards  deposited  25  %  of  the  remainder 
of  his  money ;  how  much  money  had  he  then  in  the  bank, 
and  what  per  cent  was  this  of  all  his  money  ? 

21.  A  rectangular  field  contains  110  acres,  and  37-^%  of 
the  length  is  381.183  yards  ;  what  is  the  breadth  in  rods  ? 


PERCENTAGE.  215 

22.  A  and  B  gain  in  business  $5040,  of  which  A  is  to 
have  10%  more  than  B  ;  what  is  the  share  of  each  ? 

23.  A  commission  merchant  sold  350  barrels  of  flour, 
charging  2^%  commission  and  2%  for  guaranteeing  pay- 
ment ;  the  net  proceeds  amounted  to  $2511.25 ;  find  the 
price  of  the  flour  per  barrel. 

24.  My  agent  sells  for  me  2000  yards  of  cloth  at  24  cents 
a  yard ;  he  allows  the  purchaser  5%  discount  for  cash,  and 
charges  me  2^%  on  the  cash  receipts;  how  much  money 
does  he  pay  over  to  me  ? 

25.  At  $4  a  ton,  how  many  tons  of  coal  can  be  bought 
for  f  8526,  after  paying  a  commission  of  1^%  ? 

26.  For  v/hat  sum  must  a  building  valued  at  $31200  be 
insured  so  that,  in  case  of  fire,  the  owner  may  recover  both 
the  value  of  the  building  and  the  premium  of  2^%  ? 

27.  A  jeweller  pays  $14.80  duty  on  an  imported  watch, 
which  is  20%  ad  valorem ;  what  must  he  sell  the  watch  for 
in  order  to  gain  25%  ? 

28.  A  town  is  taxed  $3600.  The  real  estate  of  the  town 
is  valued  at  $560000  and  the  personal  property  at  $152500. 
There  are  600  polls,  each  of  which  is  taxed  $1.25.  What  is 
the  tax  on  $1000  ?  What  is  A's  tax,  who  pays  for  four 
polls,  and  owns  real  estate  valued  at  $4100  and  personal 
property  valued  at  $1800  ? 


216  ARITHMETIC. 

CHAPTER  X. 

INTEREST  AND  DISCOUNT. 

126.  Money  paid  for  the  use  of  money  is  called  interest. 
The  sum  of  money  for  the  use  of  which  interest  is  paid  is 
called  the  principal,  and  the  sum  of  the  principal  and  inter- 
est is  called  the  amount.  Interest  reckoned  on  the  princi- 
pal alone  is  called  simple  interest. 

The  rate  of  interest  is  the  number  of  hundredths  of  the 
principal  taken  as  the  interest  for  one  year.  The  legal  rate 
of  interest  varies  in  different  states,  but  6%  is  the  most 
common.  In  this  book  6%  is  to  be  understood  when  no 
rate  of  interest  is  mentioned. 

In  computing  interest  it  is  customary  to  consider  a  year 
as  consisting  of  12  months  of  30  days  each.  At  6%  the 
interest  on  $1  for  one  year  is  6  cents  ;  for  one  month  it  is 
^  of  6  cents,  or  5  mills ;  for  one  day  it  is  -j^  of  5  mills,  or 
■  1  of  a  mill.  Hence,  to  find  the  interest  on  $1  for  any  given 
time  at  6%,  take  six  times  as  many  cents  as  there  are  years, 
Jive  times  as  many  mills  as  there  are  months,  and  one  sixth  as 
many  mills  as  there  are  days,  and  find  their  sum. 

I.   Find  the  interest  of  f  237.72  for  3  yr.  9  mo.  23  da.  at  6%. 

3  yr. .  .  .  .18  237.72 

9  mo.  .  .  .045  .2281 

23  da.    .  .  .003f  11886  ^he  interest  on  |1  for  3  yr.  9 

.228^  7924  °^^-  ^^  ^^-  ^^  found  by  the  method 

^         190176  l^st  given  to  be  $0.2285.    r^^^  jn- 

47544  terest  on  $237.72  is  237.72  times 

47544  $0.228|,  which  equals  $54.40. 

Ans.  $54.40.         54.39826 


ESTTEREST   AND   DISCOUNT.  217 

Note.  In  computing  interest  the  only  fractions  that  can  occur  are 
h  h  h  h  ^"^  6-  ^^^^  ^^^'  ^^y  *o  multiply  by  j  is  to  set  down  J  of  the 
multiplicand  twice ;  to  multiply  by  f ,  set  down  ^  and  J  of  the  multi- 
plicand. 

After  finding  the  interest  at  6%,  the  interest  at  any  other 
rate  can  easily  be  found  by  dividing  by  6,  and  then  multi- 
plying by  the  number  expressing  the  rate.  The  interest  at 
certain  rates  can  be  found  more  briefly  as  follows : 

8%  .  .  .  divide  the  interest  at  6%  by  2. 
4%  .  .  .  subtract  from  the  interest  at  6%  J  of  itself. 
4|-%  .  .  subtract  from  the  interest  at  6%  J  of  itself. 
5%  .  .  .  subtract  from  the  interest  at  6%  ^  of  itself. 
5^%  .  .  subtract  from  the  interest  at  G%  tj  ^^  itself. 
6^%  .  .  add  to  the  interest  at  6%  ^  of  itself. 
7%  .  .  .  add  to  the  interest  at  6%  |^  of  itself. 
7i%  .  .  add  to  the  interest  at  6%  J  of  itself. 
8%  .  .  .  add  to  the  interest  at  6%  ^  oi  itself. 
9%  .  .  .  add  to  the  interest  at  6%  ^  of  itself. 

II.  Find  the  amount  of  |1345.o0  from  Oct.  16th,  1883 
to  May  14th,  1888  at  4^%. 


By  the  method  shown  in  §  68 
we  find  the  difference  between 
the  dates  to  be  4  yr.  6  mo.  28 
da.  The  interest  of  .$1345.50 
for  this  time  at  6%  is  $369.56. 
Subtracting  from  this  result 
I  of  itself,  we  obtain  $277.17 
as  the  interest  at  4|%.  The 
amount  equals  the  sum  of  the 
principal  and  interest,  which 
$1622.67  ^^  $1622.67. 


1888—  5  —  14 

1345.50 

1883—10—16 

.2741 

4_  6—28 

44850 

44850 

4  yr.   .  .  .24 

538200 

6  mo 03 

941850 

2^  da.  .  .  .004| 

269100 

274| 

4)369.56400 

92.391 

277.17 

1345.50 

218  ARITHMETIC. 

Note.  In  computing  interest  on  English  money,  the  principal  should 
be  expressed  in  pounds  and  decimal  of  a  pound ;  the  process  then  is 
exactly  the  same  as  with  United  States  money.  The  decimal  of  the 
result  should  be  reduced  to  lower  denominations. 


EXAMPLES. 

Eind  the  interest  of 

1.  f  1000  for  1  yr.  2  mo.  12  da.  at  6%. 

2.  $257  for  3  yr.  7  mo.  24  da.  at  6%. 

3.  $237.28  for  7  yr.  2  mo.  9  da.  at  6%. 

4.  1178.99  for  3  yr.  11  mo.  14  da.  at  6%. 

5.  $1563.45  for  4  yr.  11  mo.  1  da.  at  6%. 

6.  $1000  for  6  yr.  4  mo.  15  da.  at  8%. 

7.  $1000  for  5  yr.  4  mo.  15  da.  at.  20%. 

8.  $1385.50  for  23  da.  at  7%. 

9.  $1461.75  for  4  yr.  9  mo.  at  8%. 

10.  $240  for  5  yr.  4  mo.  at  7%. 

11.  $850  for  2  yr.  5  mo.  20  da.  at  4%. 

12.  $1584  for  1  yr.  2  mo.  20  da.  at  7%. 

13.  $375.75  for  4  yr.  5  mo.  25  da.  at  ^%.  ■ 

14.  $175  for  2  yr.  7  mo.  17  da.  at  5%. 

15.  $312.17  for  5  yr.  5  mo.  5  da.  at  7%. 

16.  $206264.80  for  7  mo.  7  da.  at  3%. 

17.  $90.25  from  Feb.  7tli,  1884  to  May  11th,  1887  at  6%. 

18.  $76.72  from  Apr.  18th,  1882  to  Jan.  26th,  1885  at  6%. 

19.  $196.54  from  Aug.  15th,  1872  to  May  12th,  1880 
at  6%. 


INTEREST  AND  DISCOUNT.  219 

20.  ^15.82  from  Dec.  26th,  1881  to  July  2iid,  1882  at  7f  %. 

21.  125  from  Nov.  10th,  1884  to  July  1st,  1887  at  5%. 

22.  $64.50  from  June  25th,  1885  to  Aug.  10, 1887  at  6J%. 

23.  $257.81  from  Jan.  3rd,  1883  to  Apr.  6th,  1883  at  8%. 

24.  $580  from  May  16th,  1882  to  Oct.  8th,  1883  at  5%. 

25.  $875.26   from  Oct.  10th,  1876  to   July  10th,   1878 
at  7|%. 

26.  £lfor  Imo.  at  5%. 

27.  £17  8s.  3d.  for  2  yr.  6  mo.  17  da.  at  7^%. 

28.  £757  17s.  6d.  for  1  yr.  3  mo.  10  da.  at  4^%. 

Find  the  amount  of 

29.  $333.33  for  3  yr.  3  mo.  3  da.  at  3%. 

30.  $369.29  for  2  yr.  3  mo.  1  da.  at  9%. 

31.  $547.63  for  5  yr.  5  mo.  25  da.  at  ^%. 

32.  $762  for  3  yr.  4  mo.  26  da.  at  4J%. 

33.  $647.21  for  4  yr.  11  mo.  11  da.  at  3^%. 

34.  $392.10  for  6  yr.  9  mo.  15  da.  at  3f  %. 

35.  $175  from  Apr.  1st,  1882  to  Jan.  15th,  1883  at  5^%. 

36.  $2368  from  Dec- 19th,  1879  to  June  18th,  1885  at  6^%. 

37.  $96.52  from  Nov.  25th,  1887  to  Mar.  1st,  1888  at  4%. 

38.  $842.70  from  Mar.  5th,  1880  to  Aug.  2nd,  1887  at  5i%. 

39.  £22  10  s.  for  3  yr.  10  mo.  at  31%. 

40.  £50  12s.  5d.  for  5  yr.  2  mo.  3  da.  at  8%. 

41.  £46  6s.  8  d.  from  June  20th,  1868  to  May  5th,  1875 
at  4%. 


220  ARITHMETIC. 

127.  When  the  time  is  short,  bankers  and  business  men 
generally  compute  interest  for  the  exact  number  of  days, 
allowing  360  days  to  the  year. 

I.  Find  the  interest  of  $175.50  from  Apr.  28th  to  Aug. 
10th  at  7%. 

2  175.50  By  the  method  shown  in  §  68  we 

31  104  find  the  exact  number  of  days  to  be 

30  7Q200  104.     The  interest  of  $1  for  1  day 

31  17550  at  6%  is  i  of  a  mill,  or  ^^  of  a 
10                6000")  18252  00    -       dollar;  the  interest  of  any  princi- 

104  da.  6)30^ —  ^^^  ^^  ^^  ^^  *^^  principal.   Hence 

— the  interest  for  104  days  is  ^^  of 

•^^*  the  principal,  which  is  found  by 

3.549  multiplying  by   104   and   dividing 

Ans.   $3.55,  by  6000.    We  thus  find  the  interest 

at  6%  to  be  $3.04.    Adding  to  this 
result  i  of  itself,  we  obtain  $3.55  as  the  interest  at  7%. 

This  method  may  be  stated  as  follows :  To  find  the  in- 
terest on  any  principal  at  6%,  multiply  the  principal  by  the 
number  of  days,  and  divide  by  6000. 


EXAMPLES. 

Find  the  interest  of 

1.  $421  for  62  da.  at  6%. 

2.  $67.42  for  37  da.  at  6%. 

3.  $104.25  for  90  da.  at  7i%. 

4.  $532.50  for  47  da.  at  51%. 

5.  $1332  from  June  15th  to  Oct.  24th  at  4^%. 

6.  $2375  from  Jan.  1st  to  Mar.  20th  at  8%. 

7.  $542.36  from  Mar.  29th  to  Nov.  9th  at  5%. 

8.  $36.75  from  Apr.  12th  to  Aug.  2nd  at  6i%. 


INTEREST   AND  DISCOUNT.  221 

9.   ^212.91  from  June  30th  to  Dec.  1st  at  5J%. 

10.  $371.14  from  Nov.  8tli,  1886  to  Mar.  16th,  1887  at 
4J%. 

11.  $614.25  from  Dec.  16th,  1887  to  Apr.  25th,  1888  at 

12.  $74.16  from  Sept.  7th,  1888  to  Jan.  10th,  1889  at 
6i%. 

Exact  Interest. 

128.  The  common  methods  used  in  computing  interest 
are  based  upon  the  supposition  that  there  are  360  days  in  a 
year ;  this,  however,  does  not  give  strictly  accurate  results. 
To  obtain  the  exact  interest,  we  must  reckon  365  days  in  a 
year,  a  method  used  by  the  United  States  Government  and 
by  some  bankers  and  business  men. 

I.  Find  the  exact  interest  of  $834.32  from  Aug.  7th, 
1884  to  Jan.  12th,  1888  at  5%. 

24  834.32  41.716  125.148 

30  .05  158  18.057 

31 

30 

31 
_12 
158 

3yr.  158  da. 


2878 
2555 

The  exact  time  from  Aug.  7th,  1884  to  Jan.  12th,  1888  is  3  yr.  158  da. 
The  interest  of  $834.32  for  1  yr.  at  5%  is  $41,716;  for  3yr.  it  is  3 
times  141.716,  or  $125,148.  For  158  da.  the  interest  is  if|  of  $41,716, 
which  equals  $18,057.  Adding  these  two  results,  we  find  the  interest 
fpr  the  entire  time  to  be  $143.21. 


41.7160 
3 
X25.148 

333728       ffi-i  ^o  oi 
208580       ^^^'"^^ 
41716 
365)6591.128(18.057 
365 

2941 
2920 

2112 
1825 

222  AEITHMETIG. 

Note.  The  exact  interest  for  360  da.  is  f f  ?,  or  ff ,  of  the  interest  for 
a  year,  whereas  by  the  common  methods  the  interest  for  360  da.  equals 
the  interest  for  a  year;  hence,  for  any  number  of  days  less  than  a  year, 
the  exact  interest  is  ^^  of  the  common  interest,  and  the  exact  interest 
can  be  found  by  deducting  from  the  common  interest  yL  of  itself. 

EXAMPLES. 
Find  the  exact  interest  of 

1.  fl642for95da.  at6%. 

2.  1805  for  163  da.  at  7%. 

3.  1222.50  for  208  da.  at  4i%. 

4.  $347.75  from  Jan.  9th  to  May  26th  at  71%. 

5.  $973.96  from  Mar.  18th  to  Aug.  30th  at  31%. 

6.  1335.46  from  June  15th  to  Oct.  31st  at  7y%%. 
•       7.    $581.38  from  Apr.  19th  to  Dec.  4th  at  3%. 

8.  $767.25  from  Nov.  17th,  1885  to  Eeb.  10th,  1887  at  6  %. 

9.  $91.80  from  June  1st,  1886  to  Sept.  4th,  1888  at  4%. 
10.    $504.42  from  Dec.  1 7th,  1883  to  May  9th,  1887  at  3^% . 

To  Find  the  Rate  Per  Cent  when  the  Principal, 
Interest  (or  Amount),  and  Time  are  Given. 

129.  I.  Find  the  rate  per  cent  when  the  interest  of  $160 
for  2  yr.  3  mo.  9  da.  is  $19.11. 

2  yr.   .  .  .12  160 

3  mo 015  .1361 


9  da.    ...0011  6)21.840               The  interest  of  $160  for  2  yr. 

Toa^  3.64            ^  "^°-  ^  ^^'  ^*  ^%  ^^  $3.64.      To 

2  produce  an  interest  of  $19.11,  the 

ioo^A  ^^"^                       l%as  $3.64  is  contained  times  in 


1820 
364     4 


$19.11,  which  equals  6^%. 


INTEREST  AND   DISCOUNT.  223 

Note.  "When  the  amount  is  given,  the  interest  can  be  found  by  sulv 
tracting  the  principal  from  the  amount. 

EXAMPLES. 

1.  At  wnat  rate  per  cent  must  $370  be  put  on  interest 
to  gain  $55.50  in  3  yr.  ? 

2.  If  $50  gain  $5.60  in  3  yr.  6  mo.,  what  is  the  rate  pe? 
cent? 

3.  At  what  rate  per  cent  will  $350  amount  to  $423.5(( 
in  5  yr.  3  mo.  ? 

4.  At  what  per  cent  must  $750  be  loaned  to  amount  to 
$876  in  2  yr.  4  mo.  24  da.  ? 

5.  At  what  rate  per  cent  will  $240  in  5  yr.  give  $64 

interest  ? 

6.  The  interest  on  $325.72  for  2  yr.  4  mo.  is  $34.20; 
what  is  the  rate  per  cent  ? 

7.  At  what  rate  per  cent  will  $2500  amount  in  3  yr.  to 
$4320  ? 

8.  The  interest  on  $437.21  for  9yr.  9  mo.  is  $127,884  j 
what  is  the  rate  per  cent  ? 

9.  At  what  rate  per  cent  will  $850  earn  $97.18  in  1  yr. 
7  mo.  18  da.  ? 

10.  What  is  the  rate  per  cent' when  $320  gains  $5.28 
from  Aug.  24th  to  Oct.  30th  ? 

11.  The  interest  of  $720  from  Feb.  1st,  1886  to  Apr.  13th, 
1887  was  $69.12 ;  what  was  the  rate  per  cent  ? 

12.  The  interest  on  $127.50  from  June  26th,  1798  to  May 
8th,  1802  was  $36.975 ;  calculate  the  rate  per  cent. 

13.  At  what  rate  per  cent  will  a  sum  of  money  double 
itself  in  6  yr.  ? 


224 


ARITHMETIC. 


14.  At  what  rate  per  cent  will  a  sum  of  money  double 
itself  in  12  yr.  6  mo.  ? 

15.  What  is  the  rate  per  cent  when  a  sum  of  money  earns 
a  sum  I  as  large  in  7  yr.  2  mo.  ? 

16.  A  person  borrows  $500  on  Apr.  10th,  and  on  June 
22nd  pays  his  debt  with  $510.20.  At  what  rate  per  cent 
was  he  charged  interest  ? 

17.  If  I  buy  a  house  for  $5620  and  receive  $1803  for 
rent  in  2  yr.  3  mo.  15  da.,  what  rate  of  interest  do  I  get  for 
my  money  ? 

18.  The  interest  on  £50  12  s.  6d.  for  a  year  is  £1  15  s. 
5J  d. ;  what  is  the  rate  per  cent  ? 


To  Find  the  Time  when  the  Principal,  Interest 
(or  Amount),  and  K-ate  Per  Cent  are  Given. 

130.   I.   Pind  the  time  in  which  the  interest  of  $446  at 
5%  will  amount  to  $46.75. 


The  interest  of  $446  for  1 
yr.  at  5%  is  $22.30.  To  pro- 
duce an  interest  of  $46.75 
requires  as  many  years  as 
$22.30  is  contained  times  in 
$46.75,  which  equals  2.096 
yr.,  or  2  yr.  1  mo.  5  da. 


446 
.05 

2.096  yr. 
12 

22.30)46.75(2.096 
4460 

1.152  mo. 
30 

21500 
20070 

456  da. 

14300 
13380 

Ans.  2  yr.  1  mo.  5  da. 


Note.  When  there  is  a  fraction  of  a  day,  less  than  half  a  day  should 
be  rejected ;  half  a  day  or  more  should  be  called  another  day. 


EXAMPLES. 


1.   How  long  must  $180  be  on  interest  to  gain  $99  at 


INTEREST   AND  DISCOUNT.  225 

2.  How  long  must  f  133  be  on  interest  at  7%  to  gain 
^32.585? 

3.  In  what  time  will  $680  at  4%  amount  to  $727.60? 

4.  How  long  must  $350  be  on  interest  at  6%  to  amount 
to  $404.25? 

5.  In  what  time  will  $260  gain  $53.30  at  6%  ? 

6.  How  long  must  $125  be  on  interest  at  7|%  to  gain 
$15? 

7.  In  what  time  at  6%  will  $240  amount  to  $720? 

8.  How  long  must  $1800  be  on  interest  at  3|%  to  gain 
$11.55? 

9.  In  what  time  will  $340  produce  $111.35  interest  at 
5%? 

10.  In  what  time  will  $4500  at  5  %  gain  $181.25  ? 

11.  How  long  must  $360  be  at  interest  at  6%  to  amount 
to  $386.70? 

12.  How  long  must  $350  be  at  interest  at  6%  to  amount 
to  $395.50? 

13.  In  what  time  will  $2275  amount  to  $2673.125  at  5%? 

14.  In  how  many  days  will  $3245  gain  $80  at  7%  ? 

15.  A  man  received  $136.75  for  the  use  of  $1820,  which 
was  6%  for  the  time  ;  what  was  the  time  ? 

16.  Calculate  the  date  at  which  a  sum  of  $450,  which 
was  put  at  interest  at  8%  Dec.  30th,  1797,  amounted  to 
$642.30. 

17.  Calculate  the  date  at  which  a  sum  of  $234,  which 
was  put  at  interest  at  9%  Oct.  25th,  1798,  amounted  to  $351. 

18.  In  what  time  will  any  principal  double  itself  at  6%  ? 

19.  In  what  time  will  the  interest  on  a  sum  of  money  be 
I  of  the  principal  at  4|%  ? 


226  AEITHMETIC. 

20.  A  certain  sum  of  money  was  xjut  at  interest  at  9% 
Dec.  21st,  1790;  at  what  date  did  it  become  tripled? 

21.  A  certain  sum  of  money  was  put  at  interest  at  7^\% 
Oct.  30th,  1836 ;  at  what  date  did  it  become  tripled  ? 

22.  In  what  time  will  £1225  amount  to  £1417  18  s.  9d. 
at  3%  ? 

To    Find    the   Principal   when   the   Interest    (or 
Amount),  E-ate  Per  Cent,  and  Time  are  Given. 

131.   I.   Pind  the  principal  on  which  the  interest  for  3  yr. 
9  mo.  18  da.  at  6%  is  1210.90. 

3  yr.  .  .  .18        .228)210.900(^925  ,   ^^l  '""'''f,  ""MV"' 

9  mo.... 045  2052      ^  Ir'JZ         .'        ^''" 

18  da.... 003  —570  ^^•^^^-     io  Produce  an  m- 


.228  456 


terest   of   .$210.90   will  re- 
quire as  many   dollars   as 
1140  $0,228   is   contained   times 

114Q  in    $210.90,   which    equals 

$925. 

11.   Pind  the  principal  which  will  amount  to  $1500  in  1 
yr.  7  mo.  20  da.  at  4%. 


1  yr 06 

7  mo. .  .  .035 

1.0651)1500. 
9    9 

10  da 003i 

3).098i 
.032^ 
.065f 

9.59)13500.00(11407.72 
959 
3910 
3836 

7400 
6713 
6870 
6713 


1570 

The  amount  of  $1  for  1  yr.  7  mo.  20  da.  at  4%  is  $1.065f.  It  will 
require  as  many  dollars  to  amount  to  $1500  as  $1.065|  is  contained 
times  in  $1500,  which  equals  $1407,72. 


INTEREST   AND  DISCOUNT.  227 


EXAMPLES. 

1.  What  sum  invested  at  4%  will  yield  an  annual  in- 
come of  1100  ? 

2.  What  principal  at  5%  will  gain  $15  in  6  mo.  ? 

3.  What  principal  will  yield  an  interest  of  $339.20  in 
6  yr.  4  mo.  at  6%  ? 

4.  What  principal  will  yield  an  interest  of  $15.40  in  1 
jr.  3  mo.  at  7%  ? 

5.  What  principal  at  6%  will  amount  to  $3605.85  in 
16  mo.  ? 

6.  What  sum  of  money  at  5%  will  amount  to  $2375.38 
in  3  yr.  6  mo.  ? 

7.  What  is  the  sum  of  money  which  if  put  at  interest 
for  2  yr.  3  mo.  at  4%  will  amount  to  $230  ? 

8.  What  sum  of  money  will  produce  $3437.74  interest 

in  7  mo.  at  3%  ? 

9.  What  principal  will  gain  $176.25  in  2  yr.  4  mo.  6  da. 

at  5%  ? 

10.  At  4^%  what  principal  will  yield  an  interest  of  $360 
in  7  mo.  15  da.  ? 

11.  What  principal  will   gain  $30  in  60  days  at  2%  a 
month  ? 

12.  What  principal  will  yield  $1.70  interest  in  25  days 
at  6%  ? 

13.  What  principal  will  in  5  yr.  8  mo.  15  da.  at  5%  give 
$287.70  interest  ? 

14.  What  sum  of  money  put  at  interest  at  8%  for  2  yr. 
1  mo.  15  da.  will  amount  to  $842.40  ? 


228  ARITHMETIC. 

15.  What  sum  at  4%  will  amount  to  $578.88  in  1  yr.  9 
mo.  18  da.  ? 

16.  Find  the  sum  on  which  the  interest  at  9%  for  5  yr. 
1  mo.  18  da.  is  $947.10. 

17.  What  sum  of  money  put  at  interest  for  6  yr.  5  mo.  11 
da.  at  7%  will  earn  $3159.14  ? 

18.  Find  the  principal  which  will  amount  to  $2500  in  3 
yr.  3  mo.  10  da.  at  4|%. 

19.  What  principal  will  in  7  yr.  7  mo.  7  da.  amount  to 
$700  at  7%  ? 

20.  The  interest  on  a  certain  principal  from  Sept.  16th, 
1884  to  Oct.  20th,  1886  at  6%  was  $22.37;  what  was  the 
principal  ? 

21.  What  sum  of  money  put  at  interest  July  23rd,  1885 
at  6%  will  amount  to  $1842  Mar.  11th,  1887  ? 

22.  What  principal  will  produce  $125  interest  from  May 
24th,  1883  to  Nov.  5th,  1884  at  8%  ? 

Promissory  Notes. 

132.  A  promissory  note,  commonly  called  a  note,  is  a 
written  or  printed  promise  to  pay  a  specified  sum  of  money 
on  demand  or  at  a  specified  time.  The  sum  whose  payment 
is  promised  is  called  the  face  of  the  note ;  it  should  be  writ- 
ten in  words  in  the  body  of  the  note,  and  in  figures  either 
at  the  top  or  the  bottom.  The  person  signing  the  note  is 
called  the  maker ;  the  person  to  whom  the  sum  of  money  is 
payable  is  the  payee ;  the  owner  of  the  note  is  the  holder ; 
and  a  person  who  writes  his  name  on  the  back  of  the  note 
as  security  for  its  payment  is  an  indorser. 

A  negotiable  note  is  one  that  can  be  sold  or  transferred. 
To  be  negotiable  it  must  be  payable  to  the  "  bearer,"  or  to 


INTEREST   AND   DISCOUNT.  229 

the  "  order  "  of  the  payee.  A  note  should  contain  the  words 
"  value  received,"  otherwise  the  holder  may  be  required  to 
prove  that  its  value  was  received  by  the  maker. 

A  note  is  nominally  due  at  the  expiration  of  the  specified 
time ;  it  matures,  or  becomes  legally  due,  three  days  after 
the  specified  time  has  expired ;  these  three  days  are  called 
days  of  grace.  The  time  when  a  note  is  due  is  often  indi- 
cated by  writing  the  date  when  nominally  due  and  the  date 
of  maturity  with  a  line  between;  for  example,  Oct.  9/i2, 
1888. 

If  a  note  contains  the  words  "with  interest,"  it  draws 
interest  from  date ;  otherwise  it  draws  interest  from  the 
time  of  maturity  until  paid. 

A  protest  is  a  written  notice  made  by  a  notary  public  to 
the  indorsers  that  the  note  has  not  been  paid.  A  protest 
must  be  made  out  on  the  last  day  of  grace,  otherwise  the 
indorsers  are  released  from  their  obligation. 

Each  state  makes  its  own  laws  in  regard  to  all  kinds  of 
negotiable  paper,  and  any  note  is  governed  by  the  laws  of 
the  state  in  which  it  is  payable. 

The  following  are  common  forms  of  promissory  notes  : 

INDIVIDUAL  NOTE. 

I^y^^^.       Boston,  Mass.,  c^-^.  SO^,  ISSf. 

For  value  received,  C/  promise  to  pay  to 

u-at^<n^  or  order, 

'C^£,  Au/n€^€.€/  £.€^^^ee'^  ^ — — joo  Dollars, 

in  '^fe:  ^m,<pyi/Ad.  from  date.  ^^  y>     ^ 


230  ARITHMETIC. 


NOTE   PAYABLE  AT   A   BANK. 


Irf^c^^i-       Pittsburg,  Pa.,^.^^  y<J-^  188/. 

C/A'ik'^  ■c/ay.d-  after  date  t4X£.  promise  to  pay 
to  the  order  of  -^^~^— --^^  (W.  C^^^-^^  ^^^.^^.^^.^^ 
^-t'O.A^  Ai'iyn€i-tei^ jl^j>^:^d-ut. jQQ  Dollars, 

at  the  %\\t)^\xtxi\y^  Rational  gawfe. 

Value  received,  without  defalcation. 


Note.  The  law  of  Pennsylvania  requires  the  words  "  without  defalcation." 


NON-NEGOTIABLE  NOTE. 


Qle^  Q/<^ld,   Ocl  y^// 


'^n^  t^e'T^'C^'^t^^  Cy  fiyio^'^'T^'C-d^^  ^ 


C^^^^  J2^  Qfo^v^i^  C/^/^ 


/^. 


^e^M-e   ^^eoei^-ue^/ ,  ^iij^^ri  i-^^Cel-e^d^c    tz^ 


iU^^   /i.-ed'    'C^'^^t   /h€4^   t^^n^'Ti.'U^r^. 


^o'^  ^. 


c9  c^TSn"  vSf-e^^lt/y    (_>ri^^j^^^i^^^^ 


INTEREST  AND  DISCOUNT.  231 


Partial  Payments. 


133.  When  partial  payments  are  made  on  notes  or  other 
obligations  bearing  interest,  allowance  is  made  for  these 
partial  payments  in  computing  the  amount  due  at  the  time 
of  settlement.  The  amounts  of  the  payments  and  their 
dates  are  written  on  the  back  of  the  obligation ;  such  pay- 
ments are  called  indorsements.  The  Supreme  Court  of  the 
United  States  has  adopted  a  rule  for  partial  payments, 
which  is  known  as 


The  United  States  Rule. 

Find  the  amount  of  the  principal  to  the  time  when  the  pay- 
ment, or  the  sum  of  the  payments,  equals  or  exceeds  the  interest 
due.  From  the  amount  subtract  the  payment^  or  the  sum  of 
the  payments^  and,  with  the  remainder  as  a  new  principal, 
proceed  as  before  to  the  time  of  settlement. 

This  rule  is  based  on  the  following  principles : 

1st,  payments  must  be  applied  first  to  the  discharge  of 
interest  due,  and  the  balance,  if  any,  toward  the  discharge 
of  the  principal. 

2nd,  unpaid  interest  must  not  be  added  to  the  principal 
to  draw  interest. 

3rd,  only  unpaid  principal  can  draw  interest. 

I.  A  note  for  $750  is  dated  July  11th,  1884  and  bears 
the  following  indorsements :  Feb.  17th,  1885,  $225 ;  Dec. 
24th,  1885,  $25;  May  19th,  1886,  $375.  What  balance  is 
due  Dec.  1st,  1886,  reckoning  interest  at  6%  ? 


282  ARITHMETIC. 

1885—   2  —  17  750. 

1884—   7  —  11  .036 


7—  6 
.036 

4500 

2250 

27.000 
750. 

1885  —  12  —  24 
1885—  2  —  17 

777. 
225. 
552. 
.0514 

The  amount  of  $750  from 
July  11th,  1884  to  Feb.  17th. 
1886  is  $777,  and  subtracting 
the   1st   payment   from  this 

. ^  amount,  we  obtain  $552  for 

1^ —    '^  92  the  2nd   principal.     The   in- 

'^^H  552  terest    of    $552    from    Feb. 

2760  17th,  1885  to  Dec.  24th,  1885 

28.244  is  $28.24,  which  is  more  than 

4  CO/. K Hq  ;-;ro  the  2nd  payment ;  hence  we 

■tQQK 2 17  OT'^i         P^^^   °^  *®    ^^^    next  date. 

^  The  amount  of  $552  from 
Feb.  17th,  1885  to  May  19th, 
1886  is  $593.58,  and  subtract- 
ing   the    sum    of    the    2nd 


-    2  184 

.075J  2760 

3864 


41.584  and  3rd  payments  from  this 

^^^-  amount,  we   obtain    $193.58 

593.58  for  the  3rd  principal.    The 

400.  amount  of  $193.58  from  May 

1886  —  12  —    1  193.58  19th,  1886  to  Dec.  1st,  1886 

1886—   5  —  19  .032  is  $199.77. 

6  —  12  38716 

.032  58074 

6.19456 
193.58 


$199.77 


Note  1.  When  the  rate  is  any  other  than  6%,  each  separate  interest 
must  be  found  at  the  given  rate. 

Note  2.  When  the  interest  is  required  on  a  principal  on  which  par- 
tial payments  have  been  made,  find  the  amount  due,  and  from  that 
amount  subtract  the  difEerence  between  the  principal  and  the  sum  of 
the  payments. 


INTEEEST  AND  DISCOUNT.  233 


EXAMPLES. 


1.  On  a  note  for  f  1400,  given  Apr.  12th,  1882,  two  pay- 
ments  were  made :  Aug.  30th,  1884,  ^400 ;  Aug.  30th,  1886, 
$600.     At  6%  interest,  what  was  due  Dec.  30th,  1887  ? 

2.  A  note  for  $1000,  dated  Jan.  1st,  1883,  and  bearing 
interest  at  6%,  is  indorsed  with  three  payments  of  $80 
each,  made  on  Jan.  1st  of  1884,  1885,  and  1886.  What  was 
due  on  the  note  at  settlement  Oct.  1st,  1886  ? 

3.  Find  what  is  due  Oct.  1st,  1888  on  a  note  for  $1750 
at  6%  interest,  dated  Dec.  13th,  1884,  with  three  payments 
indorsed,  viz.:  June  10th,  1886,  $360 j  Jan.  1st,  1887,  $40; 
Aug.  10th,  1887,  $500. 

4.  On  a  note  for  $2000,  dated  July  15th,  1885,  and  bear, 
ing  interest  at  7%,  there  was  paid  $100  May  5th,  1886,  and 
$200  Jan.  1st,  1887.     Find  what  remained  due  Aug.  1st, 

"1887. 

5.  A  note  of  $600,  dated  Aug.  10th,  1885,  had  indorse- 
ments as  follows:  Feb.  4th,  1886,  $50;  July  27th,  1886, 
$10 ;  Oct.  9th,  1886,  $75.  How  much  was  due  Dec.  15th, 
1886  at  5%  interest? 

6.  A  note  for  $1580,  dated  Oct.  19th,  1882,  is  indorsed 
Sept.  6th,  1883,  with  $640;  Jan.  30th,  1884,  with  $20;  Oct. 
9th,  1884,  with  $380.  What  balance  is  due  Feb.  3rd,  1885, 
interest  at  4^%? 

7.  A  note  for  $300,  dated  May  15th,  1878,  and  bearing 
interest  at  5%,  is  indorsed  as  follows:  Feb.  25th,  1881,  $40; 
Sept.  15th,  1882,  $25 ;  May  4th,  1883,  $150.  What  balance 
is  due  Jan.  1st,  1884  ? 

8.  On  a  note  for  $1000,  dated  Jan.  1st,  1878,  due  in  one 
year,  and  bearing  interest  at  the  rate  of  6%  from  the  date 
of  maturity,  the  following  payments  were  made :  Aug.  16th, 


234  ARITHMETIC. 

1879,  ^300;  Feb.  12th,  1880,  $200;  Oct.  3rd,  1881,  $50; 
Jan.  17th,  1882,  $19;  May  31st,  1883,  $22.  What  was  due 
Jan.  1st,  1884  ? 

9.  What  is  the  interest  at  4^%  of  $360.45  from  July 
5th,  1883  to  Nov.  4th,  1885,  allowing  a  credit  of  $75  paid 
Oct.  6th,  1884  ? 

134.  When  the  whole  period  of  time  is  not  longer  than 
one  year,  business  men  commonly  employ 

The  Merchants'  Rule. 

Find  the  amount  of  the  principal  for  the  whole  time  the  note 
is  on  interest;  find  also  the  amount  of  each  payment  from  the 
time  it  was  made  until  settlement;  from  the  amount  of  the 
principal  subtract  the  amounts  of  the  payments. 

I.  Find  the  balance  due  Mar.  1st,  1888  on  a  note  for 
$875,  given  Apr.  18th,  1887,  on  which  the  following  pay- 
ments had  been  made:  July  7th,  1887,  $360;  Oct.  28th, 

1887,  $250. 


1888-  3-  1 
1887-^  4-18 

1888-3- 
1887-7- 

•  1 

■  7 

1888-  3- 
1887-10- 

1 

28 

374.04 
255.13 

10-13 
.052^- 

7-24 
.039 

4-  3 
.020^ 

629.17 

876. 
.052^ 

360. 
.039 

250. 

.0201 

920.65 
629.17 

145f 
1750 

3240 

1080 

125 

5000 

$291.48 

4375 
45.645f 

14.040 
360. 

5.125 
250. 

875. 

374.04 

255.13 

920.65 

The  amount  of  $875  from  Apr.  18th,  1887  to  Mar.  1st,  1888  is  $920.65. 
The  amount,  of  $360  from  July  7th,  1887  to  Mar.  1st,  1888  is  $374.04, 


INTEREST  AND  DISCOUNT.  235 

and  the  amount  of  $260  from  Oct.  28th,  1887  to  Mar.  Ist,  1888  is  .$256.13. 
The  sum  of  the  amounts  of  the  payments  is  $029.17,  and  subtracting 
this  from  $920.66,  we  find  $291.48  to  be  the  balance  due. 


EXAMPLES. 

1.  On  a  note  for  $1500,  dated  Jan.  1st,  1886,  and  bear- 
ing interest  at  4^%,  there  was  paid  $550  Apr.  1st,  188G,  and 
$725  Oct.  1st,  1886.     Find  what  remained  due  Jan.  1st,  1887. 

2.  A  lends  B  $1000  Feb.  12th,  1885;  B  pays  $200  Mar. 
27th,  1885,  and  $50  Dec.  12th,  1885.  Find  what  is  due  Jan. 
18th,  1886  at  6%  interest. 

3.  Find  what  is  due  Oct.  1st,  1888  on  a  note  for  $2500 
at  9%  interest,  dated  Jan.  1st,  1888,  on  which  payments  of 
$600  each  have  been  made :  Mar.  1st,  May  1st,  and  July  1st, 

1888. 

4.  Find  the  balance  due  Sept.  1st,  1888  on  a  note  for 
$600,  given  Sept.  1st,  1887,  on  which  the  following  pay- 
ments had  been  made:  Feb.  15th,  1888,  $120;  May  24th, 
1888,  $350 ;  July  20th,  1888,  $100. 

5.  A  note  for  $1372.50,  dated  Nov.  10th,  1887,  and  bear- 
ing interest  at  7%,  is  indorsed  as  follows:  Jan.  20th,  1888, 
$321;  Mar.  29th,  1888,  $490;  June  14th,  1888,  $275.  What 
balance  is  due  Sept.  10th,  1888  ? 

6.  Payments  were  made  on  a  debt  of  $2470,  due  May 
7th,  1886,  as  follows :  June  24th,  1886,  $420 ;  Aug.  3rd,  1886, 
$345;  Oct.  20th,  1886,  $500;  Nov.  29th,  1886,  $790.  What 
was  due  Jan.  1st,  1887  at  5%  interest  ? 

7.  What  is  the  interest  at  5%  of  $722.85  from  Oct.  19th, 
1886  to  May  3rd,  3887,  allowing  a  credit  of  $500  paid  Jan. 
4th,  1887  ? 


236 


ARITHMETIC. 


Compound  Interest. 

135.  Compound  interest  is  interest  reckoned  on  both  the 
principal  and  the  unpaid  interest  added  at  regular  intervals. 
The  interest  may  be  compounded,  or  added  to  the  principal, 
annually,  semi-annually,  or  for  any  other  period  of  time 
according  to  agreement. 

I.  Find  the  compound  interest  of  $800  for  2  yr.  8  mo. 
i2  da.  at  7%. 

800 
1.07 


5600 
800 
856. 
1.07 


5992 
856 
915.92 
.042 
183184 
366368 
5)38.46864 
6.41144 
44.88008 
915.92 
960.80 
800. 


The  amount  of  $1  for  1  yr.  at  7%  is  $1.07,  and  the 
amount  of  $800  is  800  times  $1.07,  which  equals  $856. 
Taking  this  as  a  new  principal,  we  find  the  amount  at 
compound  interest  for  2  yr.  to  be  $915.92,  which  we 
take  as  the  principal  for  the  remaining  8  mo.  12  da. 
Thus  the  amount  of  $800  for  2  yr.  8  mo.  12  da.  at 
compound  interest  is  $960.80.  The  original  principal 
subtracted  from  this  amount  gives  $160.80  as  the  com- 
pound interest. 


$160.80 


Note.  When  the  interest  is  compounded  semi-annually,  the  inter- 
est must  be  found  for  each  half-year  at  one  half  the  yearly  rate,  and 
similarly  for  any  other  period  of  time.  Interest  is  compounded  annu- 
ally if  nothing  is  stated  to  the  contrary. 


INTEREST  AND  DISCOUNT.  237 

EXAMPLES. 

1.  What  is  the  compound  interest  on  $1000  for  3  yr.  at 
7%? 

2.  Find  the  amount  of  $100  at  the  end  of  3  yr.  at  4^% 
compound  interest. 

3.  To  how  much  will  $1000  amount  in  4  yr.  at  20% 
compound  interest  ? 

4.  How  much  would  $350  amount  to  in  7  yr.  at  6% 
compound  interest  ? 

5.  What  is  the  amount,  at  compound  interest,  of  $500 
for2yr.  6  mo.  at  7%? 

6.  What  is  the  compound  interest  of  $25  for  3  yr.  5  mo. 
at  6%? 

7.  Find  the  amount  of  $1000  for  2  yr.  2  mo.  12  da.  at 
6%  compound  interest. 

8.  Find  the  compound  interest  of  $200  for  2  yr.  6  mo. 
18  da.  at  4%. 

9.  What  is  the  amount  of   $5216.75  from  Jan.  21st, 
1885  to  July  3rd,  1888  at  8%  compound  interest  ? 

10.  What  is  the  compound  interest  on  £47  13  s.  6d.  for 
3  yr.  4  mo.  15  da.  at  3i%? 

11.  Find  the  compound  interest  of  $720  for  2  yr.  at  7%, 
interest  being  payable  semi-annually. 

12.  What  will  be  the  amount  of  $103  for  2  yr.  6  mo.  at 
5%,  the  interest  being  compounded  semi-annually  ? 

13.  What  is  the  amount  of  $340  at  8%  for  1  yr.  3  mo., 
the  interest  being  compounded  semi-annually  ? 

14.  What  is  the  amount  of  $450  for  1  yr.  2  mo.  18  da. 
at  6%,  interest  compounding  quarterly  ? 


238  ARITHMETIC.  • 

15.  Find  the  compound  interest  of  $122.50  from  Sept. 
1st,  1884  to  Nov.  25th,  1885  at  4%,  interest  being  payable 
quarterly. 

To  Find  the  Principal  when  the  Compound  Interest 
(or  Amount),  Eate  Per  Cent,  and  Time  are  Given. 

136.  I.  Find  the  principal  on  which  the  compound  inter* 
est  for  2  yr.  6  mo.  at  6%  is  f  108.75. 


1.06 
1.06 

.157308)108.750000(1691.32 
943848 

636 
106 

1436520 
1415772 

1.1236 
1.03 

207480 
157308 

33708 
11236 

501720 
471924  ' 

1.157308  297960 

The  compound  interest  of  $1  for  2  yr.  6  mo.  at  6%  is  $0.157308.  To 
produce  an  interest  of  $108.75  will  require  as  many  dollars  as  $0.157308 
is  contained  times  in  $108.75,  which  equals  $691.32. 

Note.  When  the  amount  is  given  in  place  of  the  interest,  the  divi- 
sor should  be  the  amount  of  $1  for  the  given  time  at  the  given  rate. 


EXAMPLES. 

1.  What  principal  will  in  2  yr.  at  5%  produce  a  com- 
pound interest  of  $350 1 

2.  Find  the  principal  on  which  the  compound  interest 
for  3  yr.  at  4%  is  $468.24. 

3.  What  sum  of  money  at  6%  compound  interest  will 
amount  to  $2703  in  1  yr.  4  mo.? 

4.  At  4:%  compound  interest  what  sum  of  money  will 
amount  in  2  yr.  to  $594.88? 


INTEREST  AND  DISCOUNT.  239 

5.  What  sum  of  money,  at  10%  compound  interest,  will 
amount  to  18651.50  in  3  yr.  ? 

6.  Find  what  principal  will  amount  to  f  1000  in  3  yr.  6 
mo.  at  3^%  compound  interest. 

7.  What  principal  will  produce  $250  interest  in  1  yr. 
8  mo.  24  da.  at  6%,  the  interest  being  compounded  semi 
annually  ? 

8.  What  principal  will  amount  to  $2000  in  1  yr.  4  mo. 
15  da.  at  4%,  the  interest  being  compounded  quarterly  ? 

Annual  Interest. 

137.  Annual  interest  is  simple  interest  reckoned  on  the 
principal  and  also  on  each  year's  interest  after  it  is  due. 

I.   Find  the  annual  interest  of  $1710  for  3  yr.  4  mo.  12 
da.  at  5%. 

The  simple  interest  of  $1710 

for  3  yr.  4  mo.  12  da.  is  1287.85. 

The  interest  due  at  the  end  of 

the  first  year  draws  interest  for 

2  yr.  4  mo.  12  da. ;  the  interest 

due   at  the  end  of  the  second 

year  draws  interest  for  1  yr.  4 

mo.  12  da.;  the  interest  due  at 

3420  35  50      the  end  of  the  third  year  draws 

fiVS45420  -246      interest  for  4  mo.  12  da. ;  the 

— K7-  rij  '  K-j  oAQ      sum  of  the  interests  of  the  yearly 

'' —  S4200        unpaid  interests  is  equivalent  to 

Zoi.oo  17100  *^®  interest  of  one  year's  inter- 

«\oi  HQQnn      ®^*  ^^^  *^^  ^^™  ®^  these  periods. 

oinrr      The  interest  of  $1710  for  1  yr. 

^-^^^^        is   $85.50,   and  the  interest  of 

17.53  $85.50  for  4  yr.  1  mo.  6  da.  is 

287.85  $17.5.3;  adding  $287.85  to  this 

$305.3?  amount,  we  find  S305.38  to  Ut 

the  entire  annual  interewki- 


3  yr 18 

2—4—12 

4  mo 02 

1_4— 12 

12  da. .  .  .002 

4—12 

.202 

4—1—  6 

.246 

1710 

.202 

1710. 

3420 

.05 

240  AEITHMETIC. 


EXAMPLES. 


1.  What  is  the  interest  for  3  yr.  on  a  debt  of  $1800  at 
6%  annual  interest? 

2.  How  much  interest  is  due  on  a  debt  of  $1500,  at  6% 
annual  interest,  at  the  end  of  3  yr.  6  mo.  ? 

3.  Find  the  annual  interest  of  $1200  for  4  yr.  3  mo.  10 
da.  at  5%. 

4.  A  note  for  $500,  with  annual  interest  at  6%,  is  due 
4  yr.  6  mo.  after  date ;  if  no  interest  has  been  paid,  what 
will  be  due  at  maturity  ? 

5.  A  note  for  $2250,  with  interest  payable  annually  at 
8%,  was  paid  3  yr.  3  mo.  18  da.  after  date,  and  no  interest 
had  been  previously  paid ;  what  was  the  amount  due  ? 

6.  A  note  was  given  May  8th,  1883  for  $625,  interest 
payable  annually  at  5%;  if  no  payment  is  made,  what  will 
be  due  Mar.  8th,  1886? 

7.  Find  the  interest  due  Dec.  20th,  1888  on  a  note  for 
$725,  dated  June  11th,  1884,  with  interest  payable  annually 
at  7%,  when  no  interest  has  been  paid. 

8.  Find  the  amount  due  Apr.  4th,  1888  on  a  note  for 
$1150,  dated  Nov.  22nd,  1883,  with  interest  payable  annually 
at  4J%,  when  no  interest  has  been  paid. 

True  Discount. 

138.   Discount  is  a  deduction  made  for  the  payment  ol  a 

debt  before  it  is  due. 

The  present  worth  of  any  sum  of  money  due  at  a  future 
time  without  interest,  is  that  sum  which  put  at  interest  for 
the  given  time^  will  amount  to  the  given  sum.     The  differ- 


INTEREST   AND   DISCOUNT.  241 

ence  between  the  given  sum  and  its  present  worth  is  called 
the  true  discount. 

I.  Find  the  present  worth  of  $800,  due  in  1  yr.  7  mo. 
24  da.,  at  5%. 

1  yr 06  1.0825)  800.0000  ($739.03 

7mo....035  75775 

24  da.  .  .  .004  42250 

6). 099  32475 

.0165  97750 

.0825  97425 
32500 

The  amount  of  $1  for  1  yr.  7  mo.  24  da.  is  $1.0825;  hence  the  pres- 
ent worth  of  .$1.0825  is  $1.  The  present  worth  of  $800  is  as  many 
times  $1  as  $1.0825  is  contained  times  in  $800,  which  equals  $739.03. 
The  process  is  the  same  as  finding  the  principal,  when  the  amount,  rate 
per  cent,  and  time  are  given,  as  shown  in  §  131. 

Note.  When  the  money  is  at  compound  interest,  the  amount  of  $1 
should  be  found  at  compound  interest. 

EXAMPLES. 

1.  Find  the  present  worth  and  discount  of  $3230,  due 
in  4  yr.  10  mo.  12  da.,  at  6%. 

2.  Find  the  present  worth  and  discount  of  $2000,  due 
in  1  yr.  8  mo.,  at  4|^%. 

3.  Find  the  present  worth  and  discount  of  $2500,  due 
in  3  mo.,  at  8%. 

4.  Find  the   present  worth  and  discount  of  $1926.94, 
due  in  8  mo.  3  da.,  at  7%. 

5.  Find  the  present  worth  and  discount  of  $1025,  due  in 
36  da.,  at  10%. 

6.  What  is  the  present  worth  of  $3471.50,  due  3  mo.  9 
da.  hence,  at  7%  ? 

7.  What  is  the  present  worth  of  $1609.30,  due  in  10 
mo.  24  da.,  when  money  is  worth  5^  ' 


242  ARITHMETIC. 

8.  Find  the  present  worth  of  a  note  for  ^313.31,  due  in 
2yr.  2  mo.  2  da.,  at  3i%. 

9.  What  is  the  present  worth  of  $10000,  due  3  yr.  hence, 
at  5%  compound  interest  ? 

10.  What  is  the  present  worth  of  $678.75,  due  3  yr.  8  mo. 
hence,  at  7%  compound  interest? 

Bank  Discount. 

139.  Bank  discount  is  a  deduction  made  by  a  bank  for 
advancing  money  on  a  note  before  it  is  due,  and  it  is  the 
interest  on  the  face  of  the  note  from  the  day  of  discount  to 
the  day  of  maturity ;  this  period  of  time  is  called  the  term 
of  discount,  and  the  rate  of  interest  is  called  the  rate  of 
discount. 

The  sum  of  money  received  for  a  note  when  it  is  dis- 
counted at  a  bank  is  called  the  proceeds  or  avails,  and  it 
equals  the  face  of  the  note  minus  the  bank  discount. 

Note.  If  no  day  of  discount  is  given,  a  note  is  understood  to  be 
discounted  on  the  day  of  its  date ;  in  such  a  case  the  term  of  discount 
equals  the  time  specified  in  the  note  plus  three  days  of  grace. 

When  the  time  a  note  has  to  run  is  designated  by  months,  the  term 
of  discount  is  determined  by  subtracting  dates ;  when  the  time  is  desig- 
nated by  days,  the  term  of  discount  is  determined  by  exact  days. 

I.  Find  the  bank  discount  on  a  note  for  f  450,  due  in  60 
days,  at  5%. 

450. 
63 


1350 

2700  The  term  of  discount  is  60  da.  +  3  da.,  which  is  63 

6000)28350         ^^'    "^^^  interest  of  $450  for  63  da.  equals  $3.94. 
6)4.725 
.7875 
$3.94 


INTEREST   AND   DISCOUNT. 


243 


II.  Find  the  proceeds  of  a  note  for  $1200,  dated  Apr. 
10th,  payable  in  90  days,  and  discounted  May  14th  at  7%. 

90  days  after  Apr.  10th 
is  July  9th ;  hence  the  note 
becomes  due  July  ®/i2. 
The  term  of  discount  is 
from  May  14th  to  July  12th, 
which  equals  59  da.  The 
interest  of  $1200  for  59 
da.  at  7%  is  .^13.77.  The 
proceeds  is  .$1200  —  ^13.77, 
which  equals  .$1186.23. 
Note.   As  the  day  of  maturity  was  not  needed  in  this  problem,  the 

term  of  discount  could  have  been  determined  as  follows:  from  Apr. 

10th  to  May  14th  is  20  da.  +  14  da.,  or  34  da.,  and  93  da.  —  34  da.  =  59  da. 

III.  A  note  for  $2020,  dated  Oct.  31st,  1887,  and  payable 
in  6  months  with  interest  at  4^%,  was  discounted  Mar.  12th, 
1888  at  6%  ;  find  the  proceeds. 


17 

1200 

1200. 

30 

59 

13.77 

12 

10800 

$1186.23 

59  da. 

6000 
6000)70800 
6)11.80 
1.967 

13.77 


2020 
.030^ 
1010 
60600 
4)61.610 
15.4025 
46.2075 
2020. 
2066.21 


1888- 
1888- 


3 
12 


2066.21  When  a  note  "  with  in- 
17.56  terest "  is  discounted,  the 
$2048.65  discount  is  computed  on 
the  amount  due  at  ma- 
turity. The  amount  of 
$2020  for  6  mo.  3  da.  is 
$2066.21. 

6  mo.  after  Oct.  31st 
is  called  Apr.  30th  ;  since 
April  has  no  31st  day, 
the  time  expires  on  the 
last  day;  hence  the  note  becomes  due  Apr. 30/|j^ay 3.  The  term  of 
discount  is  1  mo.  21  da.,  and  the  interest  of  $2066.21  for  this  time  is 
$17.56.     The  proceeds  is  $2066.21  -$17.56,  which  equals  $2048.65. 


1—21 

.008^ 

2066.21 

.008^ 
103310^ 
1652968 
17.56278^ 


EXAMPLES. 

1.   Find  the  bank  discount  on  a  note  for  $125,  due  in  3 
months,  at  5%. 


244  ARITHMETIC. 

2.  What  is  the  discount  on  a  note  for  $475,  due  in  75 
days,  discounted  at  a  bank  at  4^%? 

3.  If  you  have  a  note  for  $1000,  payable  in  60  days, 
discounted  at  a  bank  at  6%,  what  sum  will  you  receive  ? 

4.  How  much  money  should  be  received  on  a  note  of 
$1000,  payable  in  4  months,  discounted  at  a  bank  where 
the  rate  of  discount  is  6%? 

5.  A  man  buys  $800  worth  of  goods  and  gives  his  note 
for  that  sum,  payable  in  90  days.  Find  the  sum  realized 
on  the  note  if  it  is  immediately  discounted  at  a  bank 
at  6%. 

6.  If  the  rate  of  discount  is  5%,  how  much  can  be  ob- 
tained on  a  note  for  $600,  payable  in  4  months,  discounted 
at  a  bank  ? 

7.  If  the  rate  of  discount  is  5^%,  how  much  can  be  ob- 
tained on  a  note  for  $1000,  payable  in  60  days,  discounted 
at  a  bank  ? 

8.  Find  the  proceeds  of  a  note  for  $1225,  due  in  30 
days,  discounted  at  5|-%. 

9.  Find  the  proceeds  of  a  note  of  $620.25,  discounted  at 
a  bank  for  53  days. 

10.  Find  the  bank  discount  and  proceeds  of  a  note  of 
$1285^  dated  Mar.  28th,  1883,  payable  Jan.  5th,  1885,  and 
discounted  at  4%. 

11.  Find  the  proceeds  of  a  four-months'  note  for  $1350, 
discounted  15  days  after  date  at  7%. 

12.  Find  the  proceeds  of  a  note  for  $25,  dated  Aug.  17th, 
payable  in  30  days,  and  discounted  Sept.  1st  at  5%. 

13.  Find  the  proceeds  of  a  note  for  $250,  dated  July  31st, 
payable  in  4  months,  and  discounted  Sept.  15th  at  4-^%. 


INTEREST  AND  DISCOUNT.  245 

14.  A  note  for  $500,  dated  Mar.  9tli,  at  3  months,  is  dis- 
counted Apr.  11th  at  8%  ;  what  is  received  for  the  note  ? 

15.  Find  the  bank  discount  on  a  note  for  $400,  dated 
Jan.  12th,  1887,  due  in  90  days,  and  discounted  Feb.  1st  at 
6%. 

16.  Find  the  proceeds  of  a  note  for  f  384.22,  at  60  days, 
dated  July  17th,  and  discounted  Aug.  2nd. 

17.  Find  the  proceeds  of  a  note,  dated  Oct.  5th,  1886,  for 
$428.50,  payable  in  6  months,  and  discounted  Jan.  1st,  1887 
at  5%. 

.  18.  What  is  the  bank  discount  on  a  note  for  $392,  paya- 
ble in  90  days  with  interest  at  6%,  and  discounted  15  days 
after  date  at  7%? 

19.  Find  the  proceeds  of  a  note  for  $625  at  5%  interest, 
due  in  60  days,  dated  Aug.  1st,  and  discounted  Sept.  21st 
at  5%. 

20.  A  note  for  $150,  dated  June  14th,  and  payable  in  4 
months  with  interest  at  5%,  was  discounted  July  20th  at 
7%  ;  find  the  proceeds. 

21.  A  note  for  $1000,  dated  Jan.  17th,  1888,  and  payable 
in  90  days  with  interest  at  7%,  was  discounted  Mar.  1st  at 
6%  ;  find  the  proceeds. 

22.  A  note  for  $1875,  dated  Aug.  30th,  1887,  and  payable 
in  6  months  with  interest  at  6%,  was  discounted  Oct.  27th, 
1887  at  8%  ;  find  the  proceeds. 

To   Find   the   Face   of  a  Note  to  Yield  a  given 
Proceeds. 

140.  I.  Find  the  face  of  a  note  for  90  days  which,  when 
discounted  at  4%,  will  yield  $300. 


246  ARITHMETIC. 

3)  .0155  .98961)300.                                The  bank  discount  on  $1 

0051^  3          3                              for  90  days  is  the  same  as 

01031  2.969      ) 900.000 (1303.13  the   interest   for  93    days, 

^  g907                            which  equals  $0.01031,  and 

1  0000  Qono                       *h^  proceeds  of  $1  equals 

01034  8907                      $0.9896|.      To   produce   a 

!98964  "3930                ^""''^'  °^  ^  ,"'"  "" 
oqpq                    qmre  as  many  dollars  as 


9610 


$0.9896f  is  contained  times 
in  $300,  which  equals 
$303.13. 


EXAMPLES. 

1.  What  must  be  the  face  of  a  note  having  4  months  to 
run  that  it  may  yield  $1959  when  discounted  ? 

2.  What  must  be  the  face  of  a  note  which,  when  dis- 
counted at  a  bank  for  60  days  at  6%,  shall  give  as  its  pro- 
ceeds $500  ? 

3.  For  what  sum  must  a  note  be  drawn  at  90  days  to 
net  $2050  when  discounted  at  7%? 

4.  Find  the  face  of  a  note  at  2  months  that  would  real- 
ize $4500  when  discounted  at  a  bank,  interest  being  6%. 

5.  I  wish  to  borrow  $560  at  a  bank ;  for  what  sum  must 
I  give  my  note  for  90  days  at  8%? 

6.  What  must  be  the  face  of  a  note  which,  discounted 
at  a  bank  for  30  days,  would  realize  $200  ? 

7.  If  the  rate  of  discount  is  5%,  for  what  amount  must 
a  note,  payable  in  4  months,  be  given  to  realize  $600  ? 

8.  Find  the  face  of  a  note  payable  in  60  days,  so  that 
the  proceeds  shall  be  $1200  when  discounted  at  5%. 

9.  For  what  amount  must  a  note,  payable  in  120  days, 
be  given  to.  a  bank  discounting  at  6%  to  obtain  $500  ? 

10.    For  what  sum  must  T  give  my  note  for  90  days  at  a 
bank,  in  order  to  receive  $1100,  money  being  worth  7%? 


INTEREST   AND  DISCOUNT.  247 


Exchange. 


141.  A  bill  of  exchange,  or  draft,  is  a  written  or  printed 
order  from  one  person  to  another  directing  the  payment  of 
a  specified  sum  of  money  to  a  third  person.  The  person 
signing  the  draft  is  called  the  drawer  ;  the  person  to  whom 
the  draft  is  addressed  is  the  drawee ;  and  the  person  to 
whom  the  money  is  payable  is  the  payee. 

A  sight  draft  is  a  draft  which  is  payable  on  presentation 
to  the  drawee.  A  time  draft  is  one  Avhich  is  payable  at  a 
specified  time  after  presentation,  or  after  date. 

When  the  drawee  accepts  a  draft,  he  writes  the  word 
"  Accepted "  with  the  date  and  his  signature ;  the  draft  is 
then  called  an  acceptance,  and  the  drawee,  who  now  be- 
comes an  acceptor,  is  responsible  for  its  payment. 

The  following  are  common  forms  of  bills  of  exchange : 
SIGHT  DRAFT. 

%^S0~.      Buffalo,  ^X.,/cm^  y^^  188/. 

O^  d-e^oA/ pay  to 

the  order  of  -^-^^^^^^^^^^-^^^(^iz-i^e^  SiXnd^^^ 

(I^a^€.   ^^i^'yit/^e^/  Ztjf/'y, Dol lars , 

value  received,  and  charge  to  the  account  of 


248  AKITHMETIC. 


TIME  DRAFT. 


$J(^^(^Wo'         Chicago,  III.,  c^^.  Ya^  188^. 

d/A^-i^  i^^^uyd  ^...v,>^^,^v.^^^^  after  date  pay  tc 
the  order  of j^ea^^^  jsC  (^d/e^^ 

c/w.^    ■^--a^ud-tZ'm/-^^^ Dollars, 

value  received,  and  charge  the  same  to  the  account  of 


FOREIGN  BILL  OF  EXCHANGE. 

Exchange  \ox£SS.  New  Yoek,  cJu^.  JO^A  188^. 

On  demand  pay  for  this  Bill  of  Exchange 
to  the  order  of C^^^^.^  J't^i^^^ 

To  ^  c;^«^««  <#«^.4  m.  c^  J 


INTEREST   AND   DISCOUNT.  249 

The  system  of  making  payments  in  distant  places  by 
transmitting  drafts  instead  of  money  is  called  exchange. 
The  business  is  commonly  carried  on  through  bankers,  who 
have  credit  in  distant  places,  and  sell  drafts  to  persons 
wishing  to  make  payments  in  those  places. 

When  a  draft  sells  for  its  face  value,  exchange  is  said  to 
be  at  par ;  when  a  draft  sells  for  more  than  its  face  value, 
exchange  is  above  par,  or  at  a  premiuin ;  when  a  draft  sells 
for  less  than  its  face  value,  exchange  is  below  par,  or  at  a 
discount. 

Domestic  or  Inland  Exchange. 

142.  Exchange  between  places  in  the  same  country  is 
called  domestic  or  inland  exchange. 

I.  Find  the  cost  of  a  draft  on  Chicago  for  $1320  when 
exchange  is  l-|-%  discount. 

1320  1320. 

Qj^i  14.85             "^^^  discount  on  .$1  is  $0.01|,  and  on 

165^  S1S05  15  $1320  it  is  $14.86.     Subtracting  this  from 

-4  QOA  '  $1320,  we  find  the  cost  of  the  draft  to  be 

^"^"^^  $1305.15. 


14.85 


II.   Find  the  cost  of  a  draft  on  Omaha  for  f  1400,  payable 
in  60  days,  when  exchange  is  J%  premium,  and  interest  5%. 

mrl?  ^^^^'  The  bank  discount  on  $1400  for  60  da. 

.0105  12.25        ^^  50/^  jg  ^j^g  gj^^g  j^g  jj^g  interest  for  63  da., 

which  equals  ^12.25,  and  the  proceeds  is 
$1387.75;  this  would  be  the  cost  of  the 
draft  if  bought  at  par.  At  ^%  the  premium 
on  $1400  is  $7 ;  adding  this  to  $1387.75, 
we  find  the  cost  of  the  draft  to  be  $1394.75. 

EXAMPLES. 

1.    What  is  the  value  of  a  sight  draft  on  Buffalo  for 
f  1800,  when  exchange  is  at  a  premium  of  1  Ji^  ? 


7000 
1400 

1387.75 

7. 

6)14.70 

2.45 

12.25 

$1394.75 

250  ARITHMETIC. 

2.  Find  the  cost  of  a  sight  draft  on  New  York  for 
$1300^  when  exchange  is  1J%  premium. 

3.  Find  the  cost  of  a  sight  draft  on  Detroit  for  $840, 
when  exchange  is  |-%  discount. 

4.  What  will  be  the  cost  of  a  draft  for  $3000,  payable 
in  30  days  after  sight,  exchange  being  1  %  discount,  and  in- 
terest 6%  ? 

5.  Find  the  cost  of  a  draft  for  $700,  payable  in  60  days, 
when  exchange  is  at  par,  and  interest  7%\        ' 

6.  What  must  be  paid  for  a  draft  of  $925,  at  60  days,  at 
6%,  exchange  being  -|%  premium  ? 

7.  What  must  be  paid  for  a  draft  of  $450  on  New  Or- 
leans, at  90  days,  exchange  being  |%  discount,  and  interest 


8.  What  will  be  the  cost  of  a  draft  for  $750,  payable  in 
60  days  after  sights  exchange  being  ^%  premium,  and  in- 
terest 7%  ? 

9.  Find  the  cost  of  a  draft  on  Baltimore  for  $1237.50, 
payable  in  30  days  after  sight,  exchange  being  -|-%  discount, 
and  interest  5%. 

To^  Find  the  Face  of  a  Draft  when  the  Cost  is 
Given. 

143.   I.  Find  the  face  of  a  sight  draft  bought  for  $3559.50, 
when  exc>>ange  is  1^%  discount. 

1. 
.01125  -^^  ^¥^0  discount  a  sight  draft 


.9887S>  ^559.50000  ($3600 
296625       • 
693250 
593250 


00 


for  $1  would  cost  $0.98875. 
$3559.50  will  buy  a  draft  for 
as  many  dollars  as  .$0.98875 
is  contained  times  in  $3559.50, 
which  equals 


INTEREST   AND  DISCOUNT.  251 

II.  What  is  the  face  of  a  draft,  payable  in  90  dayf;',  that 
can  be  bought  for  $2000,  exchange  being  1^%  premiun^,  and 
interest  4%  ? 

3). 0155  1.00461) 2000.  The  bank  discount 

005^2                  3             3             >^  on  $1  for  90  days  at 

'oios!  3.014     )6000.000($1990.71  4%i8.1i0.0103.',and  the 

.yjxyjo^                       3014  proceeds   is  $0.P896§. 

1  29860  ^^  ^^°/"  premium  the 

01034  27126  ^**^*  "^   '"^  "^""^^^  ^°^ 

Vsqfi!  "27340  ^  is  $1.0046^.    1^2000 

:Olf*  S  will  buy  a  draft  for 

as    many    dollars    as 

1.0046i  21400  $1.0046?  is  contained 

^^^^^  times  in  $2000,  which 

3020  is  $1990.71. 


EXAMPLES. 

1.  What  is  the  face  of  a  sight  draft  that  can  be  purchased 
for  $2351.70,  when  exchange  is  ^%  premium  ? 

2.  How  large  a  sight  draft  can  be  bought  for  $2500,  ex- 
change being  |%  discount  ? 

3.  What  is  the  face  of  a  sight  draft  bought  for  $1650, 
when  exchange  is  3^%  discount  ? 

4.  How  large  a  draft,  payable  in  30  days  after  sight,  can 
be  bought  for  $4018,  exchange  being  1%  premium,  and  in- 
terest 6%  ? 

5.  How  large  a  draft  on  Cincinnati,  at  par,  at  30  days, 
can  be 'bought  for  $1989,  when  money  is  worth  6%  ? 

6.  What  is  the  face  of  a  draft,  payable  in  60  days  after 
date,  that  can  be  bought  for  $2386.20,  exchange  being  1% 
premium,  and  interest  9%  ? 

7.  Find  the  face  of  a  draft  on  New  York,  at  90  days  sight, 
bought  for  $450,  exchange  at  1|%  premium,  and  interest 
5^0 


252  ARITHMETIC. 

8.  What  is  tlie  face  of  a  draft  on  St.  Paul  for  60  days 
which  may  be  bought  for  f  1000,  exchange  being  J%  dis- 
count, and  interest  7%  ? 

9.  Find  the  face  of  a  draft  on  Boston,  at  90  days  sight, 
bought  for  $75,  exchange  at*^%  premium,  and  interest  4%. 

Foreign  Exchange. 

144.  Exchange  between  places  in  different  countries  is 
called  foreign  exchange. 

Exchange  with  Europe  is  carried  on  principally  through 
large  commercial  cities,  as  London,  Paris,  Antwerp,  Geneva, 
Hamburg,  Frankfort,  Bremen,  and  Berlin. 

Sterling  Exchange,  as  exchange  with  Great  Britain  and 
Ireland  is  called,  is  quoted  at  a  certain  number  of  dollars 
per  pound  sterling. 

Exchange  with  France,  Belgium,  and  Switzerland  is 
quoted  at  a  certain  number  of  francs  per  dollar. 

Exchange  with  Germany  is  quoted  at  a  certain  number 
of  cents  per  four  reichsmarks  (marks). 

I.  Find  the  cost  of  a  bill  of  exchange  on  London  for 
£326  16  s.,  when  sterling  exchange  is  quoted  at  4.83|-. 

326.8 
4.83i 


1634  ^326  16  s.  equals  £326.8.     Since  the  value  of  £1 

9804  is  $4,831,  the  value  of  £326.8  is  826.8  times  .$4.83|, 

26144  which  equals  $1580.08. 
13072 


1580.078 
Ans.  $1580.08. 

II.   Find  the  cost  of  a  bill  of  exchange  on  Paris  for  4730 
francs,  when  Paris  exchange  is  quoted  at  5.15^. 


LNTEREST   AND   DISCOUNT.  258 


5.155)4730.000(1917.56 
46395 


9050  Since  5.156  francs   are  worth 

5155  |i,   4730    francs    are   worth    as 

38950  many  dollars  as  5.155  francs  are 

36085  contained  times  in  4730  francs, 

28650  which  equals  $917.56. 

25775 


28750 


III.    Find  the  cost  of  a  bill  of  exchange  on  Berlin  for 
2760  J  narks,  when  German  exchange  is  quoted  at  96^. 

2760 
.244 

o.^  Sjnce  .$0.06 J  is  the  Taliie  of  4  marks,  the  value  of 

11040  ^  "^^^^  ^^  ^  °^  $0,961,  or  .$0.24J.     The  value  of  2760 

5520  marks  is  2760  times  I0.24J,  wliich  equals  f666.85. 


$665.85 


The  face  of  a  bill  of  exchange  can  be  found  when  the 
cost  is  given  by  performing  the  reverse  operation  to  that 
used  in  finding  the  cost  when  the  face  is  given. 

EXAMPLES. 

1.  What  must  be  paid  in  New  York  for  a  bill  of  ex- 
change on  London  for  £725  10  s.,  when  sterling  exchange  is 
quoted  at  4.87:|  ? 

2.  Find  the  cost  of  a  bill  of  exchange  on  Dublin  for 
£296  8  s.  6d.,  when  sterling  exchange  is  quoted  at  4.85^. 

3.  What  is  the  cost  of  a  bill  of  exchange  on  Liverpool 
for  £137  15s.  4d.,  exchange  at  4.86  ? 

4.  Hov/  large  a  bill  of  exchange  on  Edinburgh  can  be 
bought  for  $2500,  when  sterling  exchange  is  quoted  at  4.87  ? 

5.  When  exchange  on  London  is  quoted  at  4.85,  what 
will  be  the  face  of  a  draft  that  can  be  bought  for  $3889.70  ? 


254  ARITHMETIC. 

6.  What  is  the  face  of  a  bill  of  exchange  on  Liverpool 
for  which  $4800  was  paid,  exchange  at  4.84|  ? 

7.  What  is  the  cost  of  a  bill  of  exchange  on  Paris  foi 
975  francs,  exchange  at  5.16  ? 

8.  Find  the  cost  of  a  bill  of  exchange  on  Geneva  for 
1822  francs,  exchange  at  5.17. 

9.  Find  the  cost  of  a  bill  of  exchange  on  Antwerp  for 
2025.25  francs,  exchange  at  5.17:|-. 

10.  What  is  the  face  of  a  bill  of  exchange  on  Paris 
bought  for  $2240.25,  when  Paris  exchange  is  quoted  at 
5.15  ? 

11.  How  large  a  bill  of  exchange  on  Geneva  can  be  bought 
for  $850,  exchange  at  5.16^  ? 

12.  When  exchange  on  Paris  is  quoted  at  5.21:|^,  what 
will  be  the  face  of  a  draft  that  can  be  bought  for  $2046.50  ? 

13.  How  much  must  be  paid  in  Boston  for  a  bill  of  ex- 
change on  Hamburg  for  2672  marks,  exchange  at  95  ? 

14.  Find  the  cost  of  a  bill  of  exchange  on  Bremen  for 
1685.25  marks,  exchange  at  94|-. 

15.  Find  the  cost  of  a  bill  of  exchange  on  Berlin  for  1050 
marks,  exchange  at  95|. 

16.  When  exchange  on  Frankfort  is  quoted  at  95^,  what 
will  be  the  face  of  a  draft  that  can  be  bought  for  $2871.40  ? 

17.  Find  the  face  of  a  bill  of  exchange  on  Hamburg  cost- 
ing $892.76,  when  German  exchange  is  quoted  at  95^. 

18.  How  large  a  bill  of  exchange  on  Berlin  can  be  bought 
for  $1500,  exchange  at  96|  ? 

Equation  of  IPayments. 

145.  Equation  of  payments  is  the  process  of  finding  the 
time  when  several  payments  due  at  different  times  can  all 


INTEREST  AND  DISCOUNT.  256 

be  paid  at  once  without  loss  to  either  debtor  or  creditor. 
This  time  is  called  the  equated  time. 

I.  A  man  owes  $1200,  of  which  $600  is  due  in  3  months, 
$400  in  5  months,  and  $200  in  6  months ;  find  the  equated 
time  of  payment. 

600  X  3  =  1800  The  use  of  $600  for  3  mo.  is  equivalent 

400  X  5  =  2000  to  the  use  of  $1   for  1800  mo. ;   tlie  use 

200  X  6  =  1200     •  of  $400  for  5  mo.  is  equivalent  to  the  use  of 

1200  )5000  ^^  for  2000  mo.;  and  the  use  of  $200  for 

6  mo.  is  equivalent  to  the  use  of  $1  for 

1200  mo.     This  amounts  to  the  use  of  $1 

for  5000  mo.,  which  is  equivalent  to   the 

use  of  $1200  for  ^^W  ^^  ^^^^  ™**''  which  equals  4^  mo.,  or  4  mo.  6  da. 


4|-  mo. 
=  4  mo.  5  da. 


II.  A  merchant  bought  the  following  bills  of  goods : 
Jan.  15th,  $600  on  2  months'  credit ;  Feb.  1st,  $300  on  3 
months'  credit ;  Mar.  25th,  $550  on  30  days'  credit ;  and 
Apr.  8th,  $400  on  60  days'  credit.  Find  the  equated  time 
of  payment. 

Mar.  15.     600  X    0  =  0  Write  the  dates  on  which 

May  1.        300  X  47  =  14100  the  several  payments   are 

Apr.  24.      550  X  40  =  22000  due  with  the  amounts  op- 

June  7.       400  X  84  =  33600  posite.    Take  the   earliest 

1850  )69700(37.7  date.  Mar.  15th,  as  a  6on- 

5550  venient  date  from  which  to 

14200  reckon   (sometimes   called 

Ans.  Apr.  22nd.        12950  ^'^"^  ^«^^)-     The  periods 

^ornrj  of    time,  reckoning    from 

Mar.  16th,  are  0,  47,  40, 
and  84  days  respectively.  Computing  as  in  the  previous  example,  we 
find  the  equated  time  to  be  38  days  after  Mar.  15th,  which  is  Apr.  22nd. 

III.  A  man  owes  $2000  due  in  8  months ;  he  pays  $500 
in  2  months  and  $800  in  3  months ;  when  in  equity  should 
he  pay  the  balance  ? 


256  ARITHMETIC. 

500  X  6  =  3000  $500  paid  in  2  mo.  is  paid  6  mo,  be. 

800  X  5  =  4000  fore  it  is  due,  and  its  use  is  equivalent 


2000-1300  =  700)7000  *«  ^^^  "se  of  $1  for  3000  mo. 

To  mo    P^^^  ^"  ^  ™^'  ^^  P^^*^  ^  "^°"  ^^f"''^  ^t  is 

due,  and  its  use  is  equivalent  to  the 

use  of  $1  for  4000  mo. ;  tliis  amounts  to  the  use  of  $1  for  7000  mo.    To 

offset  these  payments  made  before  maturity,  the  balance  of  $700  can 

be  retained  after  maturity  for  -^  of  7000  mo.,  which  equals  10  mo. 

EXAMPLES. 

1.  A  man  owes  $300  due  in  4  months,  and  $600  due  in 
7  months ;  find  the  equated  time  of  payment. 

2.  What  is  the  equated  time  for  paying  $20  due  in  20 
days,  $60  due  in  30  days,  $40  due  in  50  days,  and  $80  due 
in  75  days  ? 

3.  A  man  buys  a  house  for  $2500,  and  agrees  to  pay  $500 
down,  and  the  rest  in  4  equal  annual  instalments;  when 
can  he  justly  pay  the  whole  at  once  ? 

4.  A  merchant  owes  $2400,  of  which  $400  is  payable  in 
6  months,  $800  in  10  months,  and  $1200  in  16  months ; 
what  is  the  equated  time  of  payment  ? 

5.  Find  the  equated  time  for  the  payment  of  $400  due 
in  30  days,  $250  due  in  60  days,  and  $200  due  in  90  days. 

6.  Of  a  debt,  ^  is  to  be  paid  in  2  months,  ^  in  6  months, 
^  in  10  months,  and  the  balance  in  a  year.  Find  at  what 
time  in  equity  the  whole  should  be  paid  if  all  the  pay- 
ments were  converted  into  one. 

7.  A  debt  is  to  be  paid  ^  down,  :^  in  6  months,  |-  in  8 
months,  and  the  balance  in  a  year ;  if  the  payments  are  all 
converted  into  one,  what  is  the  equated  time  of  payment  ? 

8.  Three  bills  are  due  as  follows  :  Sept.  5th,  $275 ;  Oct. 
1st,  $180 ;  and  Nov.  20th,  $350.  Find  the  equated  time  of 
payment. 


INTEREST  AND  DISCOUNT.  257 

9.  What  is  the  equated  time  for  the  payment  of  $170 
due  Mar.  12th,  f  250  due  Apr.  12th,  11280  due  May  17th, 
and  $325  due  June  12th  ? 

10.  I  owe  three  notes  bearing  interest  from  date  :  the 
first,  dated  Jane  1st,  1886,  is  for  $450 ;  the  second,  dated 
Dec.  17th,  1886,  is  for  $750;  the  third,  dated  Mar.  15th, 
1887,  is  for  $600.  I  wish  to  substitute  for  these  a  single 
note  for  $1800;  what  should  be  the  date  of  it  ? 

11.  A  merchant  bought  goods  on  6  months'  credit  as  fol- 
lows: Mar.  20th,  $420;  May  3rd,  $270;  and  June  12th, 
$340.  When  shall  a  note  to  settle  for  the  whole  be  made 
payable  ? 

12.  Find  the  equated  time  of  payment  for  the  following 
bills  of  merchandise  :  Oct.  10th,  1887,  $625  on  60  days' 
credit ;  Nov.  1st,  1887,  $314  on  3  months'  credit ;  and  Jan. 
4th,  1888,  $266  on  30  days'  credit. 

13.  A  merchant  bought  the  following  bills  of  goods : 
Dec.  23rd,  1887,  $428  on  90  days'  credit ;  Jan.  17th,  1888, 
$206  on  2  months'  credit ;  Feb.  3rd,  1888,  $90  on  30  days' 
credit ;  and  Feb.  8th,  1888,  $214  on  60  days'  credit.  Find 
the  equated  time  of  payment. 

14.  A  man  bought  a  horse  and  carriage  for  $500  on  6 
months'  credit ;  if  he  pays  $200  down,  when  should  he  pay 
the  balance  ? 

15.  A  man  owes  $1600  due  in  9  months ;  he  pays  $300 
in. 4  months,  $200  in  6  months,  and  $300  in  8  months; 
when  is  the  balance  due  ? 

16.  A  man  owes  $600  due  in  6  months,  $900  due  in  10 
months,  and  $1200  due  in  12  months;  at  the  end  of  8 
months  he  pays  $1800;  when  in  eo.uity  should  the  re- 
mainaer  be  paid? 


258 


ARITHMETIC. 


17.  On  a  debt  of  $5000  due  in  8  months  from  Jan.  1st, 
the  following  payments  were  made :  Apr.  1st,  $500 ;  June 
1st,  $600 ;  and  Aug.  1st,  $1000.     When  is  the  balance  due  ? 


Average  of  Accounts. 

146.  Average  of  accounts  is  the  process  of  finding  the 
time  when  the  balance  of  an  account  can  be  paid  without 
loss  to  either  debtor  or  creditor. 

I.  Find  the  equated  time  for  paying  the  balance  of  the 
following  account : 


Dr. 

B.  R. 

Harvey. 

Cr. 

1888. 

1888. 

Aug.    2 

To  Mdse.,  30  da. 

1400 

Aug.    4 

By  Draft,  60  da. 

$200 

"     29 

«      (( 

350 

"    31 

"    Cash. 

300 

Sept.    7 

«       "        2  mo. 

250 

Sept.   8 

((        « 

400 

Solution. 


Sept.  1. 
Aug.  29. 
Nov.    7. 


XXX 

3  = 
0  = 

70  = 

:      1200 

0 
= 17500 

Oct.     6.    200x38=   7600 
Aug.  31.    300  X    2=      600 
Sept.   8.    400  X  10  =   4000 

1000 
900 

18700 
12200 

900              12200 

100 

)6500 
65  da. 

Ans.  Nov.  2nd. 

Following  the  method  of  equation  of  payments,  taking  Aug.  29th 
for  the  focal  date,  we  find  the  amount  of  the  debtor  side  of  the  account 
to  be  $1000,  equivalent  to  the  use  of  $1  for  18700  days.  We  find  the 
amount  of  the  creditor  side  to  be  $900,  equivalent  to  the  use  of  $1  for 
12200  days.  The  balance  on  the  debtor  side  is  $100,  equivalent  to  the 
use  of  $1  for  6500  days,  and  the  equated  time  is  65  days  after  Aug. 
29th,  or  Not.  2nd. 

Note.  In  determining  tne  maturity  of  a  note  or  draft,  3  days  of 
grace  must  be  added  to  the  specified  time. 


INTEREST   AND   DISCOUNT. 


259 


When  the  balance  of  the  account  and  the  difference  be- 
tween the  sums  of  the  products  fall  on  the  same  side,  the 
result  is  of  the  same  nature  as  a  result  in  equation  of  pay- 
ments, and  the  equated  time  is  later  than  the  focal  date. 
When  the  balance  of  the  account  and  the  difference  between 
the  sums  of  the  products  fall  on  opposite  sides,  it  is  readily- 
seen  that  an  earlier  focal  date  could  be  taken  which  would 
give  the  same  sum  of  products  on  each  side,  and  thus  this 
date  is  the  equated  time ;  hence  the  equated  time  is  earlier 
than  the  focal  date  regularly  taken. 

The  method  may  be  stated  as  follows  : 

Write  each  item  with  its  date  of  maturity  on  the  respective 
sides  of  the  account,  and  take  as  the  focal  date  the  earliest  date 
of  maturity. 

Multiply  each  item  by  the  number  of  days  intei-vening  between 
the  focal  date  and  the  date  of  maturity,  and  find  the  sums  of 
these  products  on  each  side  of  the  account.  Divide  the  differ- 
ence between  the  sums  of  the  products  by  the  balance  of  the 
account,  and  the  quotient  is  the  number  of  days  between  the 
focal  date  and  the  equated  time. 

When  the  balance  of  the  account  and  the  difference  between 
the  sums  of  the  products  fall  on  the  same  side,  count  forward 
from  the  focal  date;  ivhen  they  fall  on  opposite  sides,  count 
backward, 

EXAMPLES. 

1.  Find  the  equated  time  for  paying  the  balance  of  the 
following  account : 

Dr.  M.  P.  Bartlett.  Cr. 


Apr.  20 
May   10 


To  Mdse.,  30  da. 


.$520 
135 


May  15 


By  Cash. 


$600 


260 


ARITHMETIC. 


2.   Find  the  time  when  a  note  for  the  balance  of  the  fol- 
lowing account  should  begin  to  draw  interest : 


Dr. 

R.  J. 

Miner. 

Cr. 

1888. 
Mar.     1 
June    8 

To  Cash. 

"    Mdse. 

11500 
235 

1888. 
May  14 
July  10 

By  Mdse. 
"  Real  Estate. 

12050 
145 

3.   Find  the  equated  time  for  the  payment  of  the  balance 
of  the  following  account : 


Dr. 

J.  H. 

Adams. 

Cr. 

1888. 

1888. 

May   10 

To  Mdse.,  30  da. 

$420 

May    4 

By  Draft,  30  da. 

$750 

June  15 

«       « 

380 

June  12 

"   Cash. 

400 

"     20 

«       « 

450 

4.   Find  the  equated  time  for  the  settlement  of  the  fol- 
lowing account ; 


Dr. 


B.  P.  Harper. 


Cr. 


1888. 

1888. 

Jan.    10 

To  Mdse. 

$672 

Jan.  28 

By  Cash. 

$475 

Eeb.     7 

"      30  da. 

428 

Apr.  10 

"  Mdse. 

462 

"      24 

«       2  mo. 

550 

May  18 

"  Cash. 

250 

5.  Find  the  face  of  a  note  that  will  balance  the  following 
account,  and  the  date  at  which  it  should  begin  to  draw 
interest : 

Cr. 


Dr. 


A.  r.  Brackett. 


1887. 

1887. 

Sept.  14 

To  Mdse.,  30  da. 

$1950 

Nov.  19 

By  Cash. 

$750 

Oct.    16 

«       3  mo. 

532 

Dec.    1 

1888. 

"  Draft,  60  da. 

1000 

"      20 

2  mo. 

1178 

Feb.    4 

"  Cash. 

600 

STOCKS.  261 


CHAPTER  XL 

STOCKS. 

147.  A  corporation  is  an  association  of  individuals  au- 
thorized by  law  to  transact  business  as  a  single  person. 
The  capital  invested  in  the  business  is  called  stock,  and  it 
is  divided  into  equal  parts  called  shares.  The  owners  of 
the  shares  are  called  stockholders,  each  of  whom  holds  a 
document  called  a  certificate  of  stock,  which  is  issued  by 
the  corporation  and  specihes  the  number  of  shares  owned. 

The  usual  value  of  a  share  is  ^100,  although  it  varies  in 
different  corporations.  In  this  book  it  will  be  regarded  as 
f  100,  unless  otherwise  stated. 

The  original  or  face  value  of  a  share  is  called  the  par 
value,  and  the  value  at  which  it  sells  is  called  the  market 
value.  When  shares  sell  for  their  face  value,  they  are  said 
to  be  at  par ;  when  they  sell  for  more  than  their  face  value, 
they  are  above  par,  or  at  a  premium ;  when  they  sell  for  less 
than  their  face  value,  they  are  below  par,  or  at  a  discount. 
The  market  value  is  quoted  at  a  certain  per  cent  of  the  par 
value.  For  example,  when  stock  is  at  par,  it  is  quoted  at 
100 ;  when  it  is  8%  above  par,  it  is  quoted  at  108 ;  when  it 
is  15%  below  par,  it  is  quoted  at  85. 

Stocks  are  generally  bought  and  sold  through  the  agency 
of  brokers,  who  receive  a  commission,  called  brokerage, 
reckoned  on  the  par  value  of  the  stock.  The  usual  rate  of 
brokerage  is  -|%,  but  other  rates  may  be  charged. 

A  dividend  is  a  sum  paid  to  stockholders  from  the  profits 
of  the  business.  An  assessment  is  a  sum  sometimes  required 
of  stockholders  to  meet  losses  or  pay  expenses.     Dividends 


262  ARITHMETIC. 

and  assessments  are  generally  reckoned  at  a  certain  per 
cent  of  the  par  value.  Dividends  are  usually  declared  an- 
nually, semi-annually,  or  quarterly,  and  the  rate  per  cent  is 
called  the  rate  of  dividend. 

Bonds  are  interest-bearing  notes  issued  by  governments 
or  corporations ;  they  are  bought  and  sold  in  the  same  man- 
ner as  shares  of  corporations.  Bonds  are  commonly  desig- 
nated according  to  the  rates  of  interest  which  they  bear. 
For  example,  Virginia  6's  are  bonds  issued  by  the  state  of 
Virginia  bearing  6%  interest. 

The  methods  of  percentage  apply  to  stocks.  The  par 
value  of  the  stock  is  the  base,  and  the  premium,  discount, 
dividend,  or  assessment  is  a  percentage  of  the  par  value. 

I.  Eind  the  cost  of  32  shares  of  railroad  stock  at  8^% 
discount. 

32 

16  At  8^%  discount  the  cost  of  one  share  is  $91  J,  and 

182  the  cost  of  32  shares  is  32  times  91^,  or  |2928. 

273 


$2928 


II.    How  much,  including  brokerage  at  ^%,  must  be  paid 
for  $15000  U.S.4's  at  llli? 
15000 


7500 

1875  If  the  brokerage  is  |%,  111|  -|- 1^,  or  lllf,  represents 

15000       the  price  paid.     lllf%  of  $16000  is  $16743.75. 
15000 
15000 


$16743.75 

III.    How  much  bank  stock  at  131|^  must  be  sold  in  order 
to  receive  $4725,  brokerage  ^%  ? 


STOCKS. 

1,3126) 4725.0000 ($3600  If  the  brokerage  is  \%,  131^  -  \,  or 

39375  131 1 ,   represents   the   price    received. 

78750  $4726  is  134%  o^  ^^e  amount  obtained 

78750  by  dividing  $4726  by  1.3126,  which  is 

7^  $3600. 

IV.  Find  the  quoted  price  of  stock  when  15  shares  cost 
$1886.25. 

1  p;M  SSA  9Pi  ^^  ^^  shares  cost  $1886.26,  one  share 

ToK    n  _    OK8  *^°***  ^'^  ^^  $1886.25,  which  is  $126.76. 

125.75  _  125f .  jj^,n^,g  t,,g  quoted  price  is  126J. 

V.  A  man  owns  $5000  of  the  stock  of  a  railroad  which 
declares  a  dividend  of  4%;  what  is  the  amount  of  his 
dividend  ? 

5000 
.04  4%  of  $6000  is  $200. 


$200.00 


VI.  How  much  8%  stock  must  be  bought  to  yield  an 
income  of  $3000  ? 

.08)3000.00  ^3QQQ  jg  go,  of  $37600. 

$37500 

Note.  In  this  book  the  brokerage  is  included  in  the  price  of  a  stock, 
unless  otherwise  stated;  hence  no  account  should  be  taken  of  broker- 
age when  it  is  not  mentioned  in  the  problem. 

EXAMPLES. 

1.  How  much  must  be  paid  for  $8500  Iowa  6's  at  112|? 

2.  What  is  the  cost  at  63^  of  stock  having  a  par  value 
of  $2800  ? 

3.  A  man  bought  28  shares  of  railroad  stock  at  18% 
discount ;  what  did  they  cost  him  ? 

4.  Find  the  cost  of  36  shares  of  bank  stock  at  121^, 
brokerage  ^%. 


264  ARITHKETIC. 

5.  How  much,  including  brokerage  at  ^%,  must  be  paid 
for  $3500  Tennessee  &s  at  88^  ? 

6.  How  much  will  be  received  from  the  sale  of  $11200 
U.  S.  3^'s  at  107f,  brokerage  ^%  ? 

7.  A  speculator  bought  45  shares  of  stock  at  4|^%  dis- 
count, and  sold  it  at  2^%  premium ;  what  was  his  gain  ? 

8.  A  broker  bought  84  shares  of  railroad  stock  at  19% 
discount.  He  sold  35  shares  at  27^%  discount,  and  the 
balance  at  8%  discount.  Did  he  gain  or  lose,  and  how 
much? 

9.  How  many  shares  of  stock  at  78%  premium  can  be 
bought  for  $9790  ? 

10.  How  much  stock  can  be  bought  for  $14178,  when  the 
quoted  price  is  208|  ? 

11.  What  amount  of  Union  Pacific  bonds  at  104|-  can  be 
bought  for  $7837.50  ? 

12.  Find  the  number  of  shares  of  bank  stock  at  105  that 
can  be  bought  for  $25260,  including  brokerage  at  ^%. 

13.  A  broker  receives  $3762.50  to  invest  in  stocks  at 
$75  per  share  and  cover  his  brokerage  at  |-%.  How  many 
shares  should  he  purchase  ? 

14.  How  much  canal  stock  must  be  sold  at  136f  in  order 
to  receive  $6552,  brokerage  i%  ? 

15.  A  broker  received  $10.50  for  selling  stock  at  122-^  j 
how  many  shares  did  he  sell,  brokerage  ■^%  ? 

16.  I  sent  $40100  to  a  broker  for  the  purchase  of  bank 
stock  at  par.  If  the  brokerage  is  :^%,  what  does  he  pay  for 
the  stock,  and  what  is  his  brokerage  ? 

17.  A  man  exchanged  72  shares  of  bank  stock  at  85  for 
railroad  stock  at  136 ;  how  many  shares  of  railroad  stock 
did  he  receive  ? 


STOCKS.  265 

18.  A  speculator  bought  stock  at  1^%  discount,  and 
gained  $495  by  selling  the  same  at  6%  premium;  how 
many  shares  did  he  purchase  ? 

19.  Bought  bonds  at  115,  and  sold  at  110,  losing  f  300. 
How  many  bonds  of  f  1000  each  did  I  buy  ? 

20.  Find  the  quoted  price  of  stock  when  35  shares  cost 
$2931.25. 

21.  What  must  be  the  quoted  price  in  order  that  $6800 
stock  may  be  bought  for  $2941  ? 

22.  How  should  U.  S.  4^'s  be  quoted  when  $10093.75  is 
paid  for  bonds  having  a  par  value  of  $8500  ? 

23.  When  the  cost  of  $4500  telegraph  stock,  including 
brokerage  at  ^%,  is  $7380,  what  is  the  quoted  price  ? 

24.  Find  the  quoted  price  of  railroad  stock  when  the 
cost  of  250  shares,  including  brokerage  at  -J^,  is  $30312.50. 

25.  Find  the  quoted  price  of  bank  stock  when  $10175  is 
received  for  110  shares,  brokerage  i%. 

26.  A  man  bought  stock  at  115,  and  sold  the  same  at 
128J ;  what  per  cent  of  the  investment  did  he  gain  ? 

27.  If  I  buy  railroad  stack  at  20%  discount,  and  sell  at 
10%  premium,  what  per  cent  do  I  gain? 

28.  A  railroad  declares  a  dividend  of  3^%  ;  how  much 
will  a  man  owning  48  shares  receive  ? 

29.  The  capital  of  a  manufacturing  company  is  $300000, 
and  it  declares  a  semi-annual  dividend  of  4%  ;  find  the  en- 
tire amount  of  the  dividend. 

30.  An  insurance  company  calls  an  assessment  of  2f  %  to 
meet  losses  ;  how  much  is  the  assessment  on  $7200  stock  ? 

31.  A  man  owns  150  shares  .of  mining  stock,  and  the 
company  declares  a  dividend  of  6%  payable  in  stock  ;  how 
many  shares  will  he  then  own  ? 


266  ARITHMETIC. 

32.  Find  the  total  income  from  $4000  9%  stocks  and 
17800  7%  stocks. 

33.  How  much  7%  stock  must  a  man  own  in  order  to 
receive  an  income  of  $4200  ? 

34.  If  a  person  receives  $360  when  a  4J%  dividend  is 
declared,  how  many  shares,  $50  each,  does  he  own  ? 

35.  The  net  earnings  of  a  corporation  are  $2625,  from 
which  a  dividend  of  6J%  is  declared ;  find  the  capital. 

36.  Mnd  the  number  of  shares  owned  by  a  person  after  re- 
ceiving 12  shares  when  a  stock  dividend  of  15%  is  declared. 

37.  Find  the  rate  of  dividend  when  a  man  owning  52 
shares  receives  $182. 

38.  A  company  with  a  capital  of  $325000  calls  an  assess- 
ment of  $4875 ;  what  is  the  rate  ?    - 

39.  A  company,  whose  capital  is  $275000,  has  $15125 
from  its  earnings  to  divide.  What  per  cent  dividend  can  it 
declare  ? 

40.  The  capital  of  a  company  is  $175000 ;  the  gross  re- 
ceipts are  $35930,  and  the  expenses  are  $19205 ;  find  the 
rate  of  dividend  that  can  be  declared  after  reserving  a  sur- 
plus of  $2725. 

148.  In  making  investments,  it  is  necessary  to  consider 
both  the  market  value  of  a  stock  and  the  rate  of  dividend. 

I.  What  income  will  be  realized  from  investing  $10650 
in  8%  stock  at  142? 

142)10650.00(7500        7500 

J)94_  -08  If  .$10650  is  invested   at  142, 

710  $600.00       the    par    value    of    the  stock  is 

710  $7500 ;  8%  of  $7600  is  $600. 

00 


STOCKS.  267 

II.  What  sum  must  be  invested  in  6%  bonds  at  92^  to 
yield  an  income  of  $1500  ? 

.06)1500.00 
25000 

•92^  To  yield   an  income  of  $1500,  the  par 

12500  value  of  the  stock  must  be  .$25000,  and  this 

50000  amount  of  stock  at  92^  will  cost  $23126. 

225000 
$23125.00 

III.  When  9%  stock  is  quoted  at  192,  what  rate  of  inter- 
est does  the  investment  pay  ? 

64)3.00(.04i^=  4|J%       The  cost  of  a  share  is  $192,  and  the  in- 
256  come  is  $9,  which  is  yf  j,  or  ^\^%,  of  the 

44      11  cost. 

64""l6 

IV.  What  is  the  quoted  price  of  a  6%  stock  which  pays 
4|%  interest  on  the  investment  ? 

.04J)6.00  The  income  is  $6,  which  is  4|%  of  the 

9 9  market  value  ;  hence  the  quoted  price  is  as 

.40)54.00  much  as  0.04|  is  contained  times  in  6,  which 

135  is  135. 

EXAMPLES.  ^ 

1.  What  annual  income  would  a  man  receive  from 
$9810  invested  in  railroad  stock  at  109,  and  paying  5^/o 
dividend  ? 

2.  What  income  will  $10120  yield  if  invested  in  4% 
bonds  bought  at  115  ? 

3.  What  income  would  a  man  receive  from  $9525  in- 
vested in  Mexican  Central  4's  at  63^  ? 

4.  How  much  will  be  realized  yearly  from  an  invest- 
ment of  $7620  in  a  5%  stock  bought  at  95,  brokerage  i%  ? 


268  ARITHMETIC. 

5.  A  man  invests  $11459  in  telephone  stock  at  204|-, 
paying  ^%  brokerage.  What  will  he  receive  when  a  divi- 
dend of  5%  is  declared  ? 

6.  How  much  must  be  invested  in  8%  stock  at  170f  to 
afford  an  income  of  |2000  ? 

7.  What  sum  must  I  invest  in  6%  bonds,  selling  at  2|-% 
premium,  to  secure  an  annual  income  of  $840  ? 

8.  How  much  mu^t  be  invested  in  a  stock  at  213|-, 
which  pays  5%  semi-annual  dividends,  to  realize  an  annual 
income  of  $420  ? 

9.  What  sum  must  be  invested  in  U.  S.  4's  at  121^, 
brokerage  ^%,  to  secure  an  annual  income  of  $700  ? 

10.  When  Wisconsin  Central  5's  are  selling  at  85|-,  how 
much  must  be  invested  to  produce  an  income  of  $750,  bro-. 
kerage  i%  ? 

11.  If  a  6%  stock  is  at  120,  what  rate  per  cent  will  an 
investor  receive  on  his  money  ? 

12.  Bank  stock,  which  sells  at  170,  pays  an  annual  divi- 
dend of  12^%  ;  what  rate  of  interest  does  a  buyer  receive  ? 

13.  Keceived  6%  dividend  on  stock  bought  at  25%  below 
par ;  what  rate  of  interest  did  the  investment  pay  ? 

14.  Stock  bought  at  20%  below  par  paid  7%  ;  what  was 
the  rate  on  the  investment  ? 

15.  What  .per  cent  of  income  does  stock  paying  8%  divi- 
dends yield,  if  bought  at  168^  ? 

16.  Which  is  the  better  investment,  a  4%  stock  at  120, 
or  a  5%  stock  at  166f  ? 

17.  Which  is  the  more  profitable  stock  to  invest  in,  3% 
at83i,  or3i%  at  97  ? 


STOCKS.  269 

18.  "Wnich  is  the  more  profitable,  88400  invested  in  5 
per  cents  at  105,  or  in  7  per  cents  at  150  ? 

19.  Which  will  yield  the  better  income,  a  4%  stock  at 
73,  or  a  7%  stock  at  126f,  brokerage  ^%  in  each  case  ? 

20.  At  what  price  must  I  purchase  8%  stock  that  the 
investment  shall  pay  5%  ? 

21.  What  must  be  paid  for  7%  bonds  that  the  invest- 
ment may  yield  6%  ? 

22.  A  bank  declares  a  semi-annual  dividend  of  4%  ;  what 
could  I  afford  to  pay  for  its  shares  if  I  wish  to  get  6%  a 
year  for  my  money  ? 

23.  At  what  price  must  a  stock  paying  6%  dividends  be 
bought  to  pay  the  same  income  as  an  8%  stock  at  par  ? 

24.  If  money  is  worth  3%,  what  is  the  premium  on  gov- 
ernment 3^%  bonds  ? 

25.  If  I  invest  $1500  in  3%  stock  at  75,  what  is  my  in- 
come, and  what  rate  per  cent  do  I  get  on  my  investment  ? 

26.  If  a  man  invests  $1338  in  bank  stock  at  167^,  what 
is  the  rate  of  dividend  when  he  receives  $120  ? 

27.  A  5%  stock  pays  a  dividend  of  $510;  if  it  is  sold 
for  $11985,  what  premium  is  paid  ? 

28.  What  must  be  the  price  of  a  5%  stock  in  order  to 
yield  the  same  rate  of  income  as  a  4%  stock  at  87  ? 

29.  When  stock  is  quoted  at  120,  what  rate  of  dividend 
must  be  paid  in  order  to  yield  the  same  rate  of  income  as  a 
6%  stock  at  144? 

30.  A  man  having  a  certain  sum  of  money  to  invest  has 
an  opportunity  of  purchasing  7%  stock  at  95,  but  delays 
until  it  has  risen  to  110.  What  per  cent  is  his  income  less 
than  if  he  purchased  at  the  first  price  ? 


270  ARITHMETIC, 

SI.  A  man  seUs  $10000  3^%  bonds  at  109^  and  rein- 
vests  the  proceeds  in  3%  bonds  at  92;  is  his  income  in- 
creasecf  or  diminished,  and  by  what  amount  ? 

32.  If  a  man  sells  $4000  6%  bonds  at  113f,  and  invests 
the  proceeds  in  4|^%  bonds  at  91,  is  his  income  increased  or 
diminished,  and  by  what  amount  ? 

33.  A  man  sold  $6000  of  6%  stock  at  144|-,  and  invested 
the  proceeds  in  8%  stock  at  170.  How  much  8%  stock  did 
he  buy,  and  what  was  the  change  in  his  income  ? 

34.  How  much  3|-%  stock  must  I  sell  at  84,  to  enable  me 
to  buy  $7700  4%  stock,  the  value  of  the  stock  being  pro- 
portional to  the  dividends  they  pay  ? 

35.  If  I  exchange  48  shares  of  a  9%  stock  at  176  for 
U.  S.  4's  at  116^,  how  much  must  I  add  to  my  investment 
to  secure  the  same  income  ? 

36.  If  I  sell  $5000  Alabama  6's  at  132  and  buy  sufficient 

U.  S.  4-^'s  at  108  to  secure  an  income  of  $225,  how  much 
shall  I  have  left,  brokerage  ^%  for  each  transaction  ? 

37.  A  man  has  a  certain  sum  of  money  to  invest.  He 
finds  that  by  buying  5%  stock  at  90  his  income  will  be  $10 
more  than  if  he  bought  8%  stock  at  150.  How  much  money 
has  he  to  invest  ? 

38.  A  man  sold  $4500  of  9%  stock  at  172^,  and  invested 
the  proceeds  in  4%  stock,  thereby  increasing  his  income  by 
$55.     Find  the  price  of  the  4%  stock. 

39.  By  selling  his  6%  stock  at  147,  and  investing  the 
proceeds  in  a  5%  stock  at  96^,  a  man  increases  his  income 
by  $54.     How  much  6%  stock  did  he  sell  ? 

40.  A  person  sells  a  certain  amount  of  5%  stock  for  86, 
and  invests  in  6%  stock  at  103,  and  by  so  doing  changes  his 
income  by  $1.  Is  the  change  an  increase  or  decrease  ?  How 
much  stock  does  he  sell  ? 


INVOLUTION  AND  EVOLUTION.  ^71 


CHAPTER   XII. 
INVOLUTION  AND  EVOLUTION. 

149.  A  power  of  a  number  is  the  number  itself,  or  the 
product  obtained  by  taking  the  number  several  times  as  a 
factor. 

A  root  of  a  number  is  one  of  the  equal  factors  of  that 
number. 

A  power  or  root  receives  its  name  from  the  number  of 
equal  factors.     For  example, 

3^  =  3.     3  is  the  first  power  of  3. 

3^  =  9.  9  is  the  second  power,  or  square,  of  3 ;  and  3  is 
the  second  root,  or  square  root,  of  9. 

3*  =  27.  27  is  the  third  power,  or  cube,  of  3 ;  and  3  is 
the  third  root,  or  cube  root,  of  27. 

3*  =  81.  81  is  the  fourth  power  of  3 ;  and  3  is  the  fourth 
root  of  81. 

The  radical  sign,  V,  indicates  a  root.  The  name  of  the 
root  is  indicated  by  a  small  figure  placed  in  the  opening  of 
the  sign,  called  the  index  of  the  root.  In  expressing  square 
root,  the  radical  sign  is  generally  used  alone.  For  example, 
V49,  or  V49,  denotes  the  square  root  of  49 ;  V256  denotes 
the  fourth  root  of  256. 

Note.  A  root  may  also  be  indicated  by  a  fractional  exponent.  For 
example,  25'  denotes  the  square  root  of  25 ;  64^  denotes  the  cube  root 
of  the  square  of  64. 

Involutiojq^. 

150.  The  process  of  finding  a  power  of  a  number  is  called 
volution. 


involution, 


272  AKITHMETIC. 

I.   Find  the  fourth  power  of  6. 

The  fourth  power  of  6  is  the  product 
6^  =6x6x6x6  =  1296.      ^f  f^^j.  factors,  each  equal  to  6,  which 

equals  1296. 

EXAMPLES. 

1.  What  is  the  square  of  11?  of  0.11? 

2.  Find  the  square  of  0.9 ;  of  three  millionths. 

3.  What  is  the  third  power  of  0.1?  of  100? 

4.  What  is  the  third  power  of  3  ?  of  0.3  ?  of  0.03  ?  of  30  ? 

5.  Find  the  cube  of  10.1;  of  1.01. 

6.  What  is  the  cube  of  |?  of  0.006  ? 

7.  What  is  the  fourth  power  of  2?  of  0.2?  of  0.02? 

8.  Find  the  fifth  power  of  5 ;  of  50 ;  of  0.5. 

Note.  The  student  will  find  it  advantageous  to  commit  to  memory 
the  squares  of  all  the  numbers  from  1  to  25  inclusive,  and  the  cubes  of 
all  the  numbers  from  1  to  10  inclusive. 

Evolution. 

151.   The  process  of  finding  a  root  of  a  number  is  called 

evolution.     When  the  exact  root  of  a  number  can  be  found, 

the  number  is  called  a  perfect  power ;  all  other  numbers 

are   imperfect  powers.     The   roots  of  perfect  powers  can 

readily  be  found  by  factoring. 

I.   Find  the  cube  root  of  9261. 

3)9261 

3)3087 

3)1029 
n\oAo  The  prime  factors  of  9261  are  S^XT^;  hence  the 

i^^      cube  root  of  9261  is  3  X  7,  or  21. 
7)49 

T 
u 

3  X  7  =  2i, 


INVOLUTION  AND  EVOLUTION.  273 

EXAMPLES. 

1.  Find  the  square  root  of  3136. 

2.  Find  the  square  root  of  5184. 

3.  Find  the  square  root  of  11025. 

4.  Find  the  cube  root  of  32768. 

5.  Find  the  cube  root  of  91125. 

6.  Find  the  cube  root  of  456533. 

7.  Find  the  fourth  root  of  331776. 

8.  Find  the  fourth  root  of  1185921. 

9.  Find  the  fifth  root  of  1889568. 
10.  Find  the  sixth  root  of  2985984. 

Square  Root. 

152.  To  obtain  a  general  method  for  finding  the  square 
root  of  numbers,  we  must  investigate  the  relations  between 
simple  numbers  and  their  squares. 

The  first  step  in  extracting  the  square  root  of  a  number 
is  to  determine  the  number  of  figures  in  the  root.  V  =  1, 
102  ^  100, 1002  ^  10000, 10002  ^  1000000,  and  so  on.  Hence 
the  square  of  any  number  between  1  and  10  is  a  number 
between  1  and  100,  the  square  of  any  number  between  10 
and  100  is  a  number  between  100  and  10000,  the  square  of 
any  number  between  100  and  1000  is  a  number  between 
10000  and  1000000,  and  so  on.  Thus  we  see  that  the  square 
of  a  number  contains  twice  as  many  figures  as  the  number, 
or  twice  as  many  less  one.  If,  therefore,  a  number  be  sep- 
arated into  periods  of  two  figures  each  by  placing  a  dot 
over  every  alternate  figure,  beginning  with  the  units'  figure, 
the  number  of  figures  in  the  root  equals  the  number  of 
periQiis. 


2T4:  ARITHMETIC. 

Note  1.  The  left-hand  period  has  but  one  figure  when  the  numbei' 
consists  of  an  odd  number  of  figures. 

Note  2.  The  principle  applies  also  to  decimals,  because  the  s  juare 
of  a  decimal  contains  twice  as  many  decimal  places  as  the  decimal 
itself. 

153.  The  component  parts  of  the  square  of  a  number  of 
two  figures  may  be  learned  from  the  following  multiplica' 
tion: 

47  =  40  +  7 

47= 40  +  7 

329  =  40  X  7  +  V 

1880  =  40^+  40x7 

472  =  2209  =  402  +  2  X  (4Q  x  7)  +  7^ 

In  general,  the  square  of  any  number  composed  of  tens 
and  units  is  equal  to  the  square  of  the  tens,  plus  twice  the 
product  of  the  tens  by  the  units,  plus  the  square  of  the  units. 

I.    Find  the  square  root  of  5329. 

5329^70  4-3  =  73  Since  the  number  consists 

4900  of  two  periods,  the  square  root 

140  -+-  S  =  143'>429  ^^^^  consist  of    two    figures. 

429  ^^^  square  of  the  tens'  figure 

of  the  root  must  be  the  largest 

^  7    .   ,T  square  in  53  hundreds,  which 

Or  more  oriefly  •  At^-u     a    a     u        *i    * 

^  ^  IS  49  hundreds ;  hence  the  tens 

5329(73  figure  of  the  root  is  7.     Sub- 

49  tracting  4900,  the  square  of  70, 

143)429  from   5329,  the   remainder  is 

,  429  429.     Since  the  square  of  the. 

tens  has  been  subtracted,  429 

equals  twice  the  product  of  the  tens  by  the  units  plus  the  square  of 

the  units ;  this  is  the  same  as  twice  the  tens  plus  the  units  multiplied 

by  the  units.     Thus  the  two  factors  of  429  are  twice  the  tens  plus  the 

units  and  the  units.     We  can  find  the  units'  figure  by  dividing  429  by 

the  other  factor;  however,  this  factor  is  known  only  in  part,  so  we  take 

twice  the  tens,  tlie  part  known,  as  a  trial  divisor,  and  we  find  that  140 

is  contained  in  429  three  times.     The  units*  figure,  therefore,  fs  3,  and 

the  complete  divisor  is  140  +  3,  or  143.    Multiplying  143  by  3,  we  obtain 


INVOLUTION   AND   EVOLUTION.  275 

429,  and  there  is  no  further  remainder.     Hence  the  square  root  of  5329 
is  73. 

In  the  shorter  arrangement  of  work,  14  may  be  considered  as  the 
trial  divisor,  and  the  units'  figure  is  found  by  dividing  42  by  14,  which 
gives  the  same  result  as  dividing  429  by  140. 

If  the  number  consists  of  more  than  two  periods,  after 
finding  the  first  two  figures  of  the  root,  we  can  consider 
them  as  tens  in  reference  to  the  next  figure,  and  then  pro- 
ceed as  before.  At  each  stage  of  the  work  the  trial  divisor 
is  obtained  by  doubling  that  part  of  the  root  already 
found. 

II.  Find  the  square  root  of  6796449. 

6796449(2607 

^  After  finding  26  in   the  root,  the  trial  di- 

4.fi^97Q  visor  is  52;  52  is  not  contained  in  30,  and  the 

(yra.  next  figure  of  the  root  is  0.     Then  the  trial 

K9n7\  divisor  is  520,  which  can  be  used  at  once  by 

bZm)   ^^^49  bringing  down  another  period. 

III.  Find  the  square  root  of  2.5  to  four  decimal  places. 

2.50(1.5811 
1 


25)150 
125 


308)2500 
2464 


3161)3600 
3161 


A  zero  must  be  annexed  to  5  to  complete 
the  first  period  after  the  decimal  point.  Other 
periods  of  two  zeros  each  can  be  brought  down 
as  they  are  needed. 


31621)43900 

The  square  root  of  a  fraction  in  its  lowest  terms  may  be 
obtained  by  taking  the  square  root  of  both  terms  when  they 
are  perfect  squares.  For  example,  the  square  root  of  ff  is 
f ;  the  square  root  of  7^  is  the  same  as  the  square  root  of 
^-,  which  equals  |,  or  2|.     When  either  term  is  not  a  per- 


276 


ARITHMETIC. 


feet  square,  reduce  the  fraction  to  a  decimal  and  then  ex> 
tract  the  root. 

When  the  denominator  of  a  fraction  is  the  square  root  of 
a  number,  the  work  may  be  simplified  by  first  multiplying 
boHh.  terms  of  the  fraction  by  the  denominato-r.     For  ex- 

EXAMPLES. 

Find  the  square  root  (to  five  decimal  places  when  the 
number  is  not  a  perfect  square)  of 


1.  676. 

2.  1681. 

3.  624100. 

4.  46656. 
6.   6.7081. 

6.  49.2804. 

7.  747.4756. 
0.005625. 
1361610000. 
4.190209. 
3444736. 


12.   0.05331481. 
37.   Find  the  value 


13.  7.333264. 

14.  39.037504.' 

15.  0.9. 

16.  0.001. 

17.  0.196. 

18.  530. 

19.  3369. 

20.  79000. 

21.  0.002539. 

22.  0.01952. 

23.  102.002. 

24.  0.001601. 


25.  AV 

27.  4. 


A- 


28. 
29. 
30. 
31. 
32. 

33.  5|. 

34.  24J. 

35.  42f 

36.  201^. 


f 

30i. 
22tV 


»'aS 


23 


.625 


to  four  decimal  places. 


38.  Find  the  value  of  V(1.06)^  to  five  decimal  places* 

39.  Find  the  value  of  VJ  to  three  decimal  places. 

40.  Find  the  value  of  —  to  three  decimal  places. 

V3 

41.  Calculate  the  value  of    \3  +  2  V2  to  two  decimal 
places.  ■ 


INVOLUTION  AND   EVOLUTION.  277 

42.  Multiply  3.15  by  0.075,  and  extract  the  square  root 
of  the  product  to  three  decimal  places. 

43.  Multiply  903.14  by  0.063,  and  extract  the  square  root 
of  the  product  to  three  decimal  places. 

44.  Divide  3.63  by  2.353,  and  find  the  square  root  of  the 
quotient  to  three  decimal  places. 

45.  Extract  the  square  root  of  0.875  -*-  2.63  to  three  deci- 
mal places. 

46.  Multiply  V2  by  V0.12S,  and  carry  the   result  to 
three  decimal  places. 

47.  The  area  of  a  square  is  655.36  sq.  ft. ;  what  is  the 
length  of  its  side  ? 

48.  If  a  square  field  contains  10.24^,  find  the  length  of 
its  side  in  meters. 

154.    Square  root  can  also  be  explained  by  the  aid  of 
diagrams. 

The  area  of  a  square  surface  is  found  by  squaring  the  length  of  one 
side;  hence  the  length  of  one  side  may  be  found  by  extracting  the 
square  root  of  the  number  denoting  the 
A  E  B  area. 

Let  ABCD  represent  a  square  contain- 
ing 676  square  units ;  we  wish  to  determine 
the  length  of  one  side. 
^         ^  Since   the   number   denoting  the   area 

consists  of  two  periods,  the   square  root 
M  N       will  consist  of  two  figures.    The  square  of 

\       h        \        c       \d\        the  tens'  figure  of  the  root  must  be  the 
P  0      largest  square  in  6  hundreds,  which  is  4 

hundreds;   hence  the  tens'  figiu-e  of  the 
676(26  root  is   2,  and   the   length  of   a   side  of 

4  the  square  is  20  plus  the  units'  figure  of 

46)276  *^^  '°^*-     ^^'  '^^  ^^  ^^  vimis  in  length; 

276  then  a  represents  a  square  whose  area  is 

400.     Subtracting  this  from  676,  we  find 

the  area  of  the  irregular  figure  EBCDFG  to  be  276.    This  figure  con- 


a 
G 

b 

c 

d 

278  ARITHMJJTIC. 

sists  of  the  two  rectangles  b  and  c  and  the  square  d;  arranging  them 
as  in  MNOP,  we  have  a  rectangle  whose  width  is  the  units'  figure  of 
the  root,  and  whose  length  is  40  plus  the  units'  figure  of  the  root. 
When  the  area  and  length  of  a  rectangle  are  known,  the  width  can  be 
found  by  dividing  the  area  by  the  length ;  in  this  case  there  is  a  small 
part  of  the  length  unknown,  so  we  take  40  as. a  trial  divisor,  and  by 
dividing  276  by  40,  we  obtain  6  for  the  units'  figure.  The  entire  length, 
then,  is  46,  and  the  breadth  is  6.  The  product  of  46  and  6  is  276,  and 
there  is  no  further  remainder.     Hence  26  is  the  square  root  of  676. 

Cube  Koot. 

155.  A  general  method  for  finding  the  cube  root  of  num- 
bers can  be  obtained  by  pursuing  lines  of  investigation 
similar  to  those  made  use  of  in  determining  the  method  for 
square  root. 

Since  13  =  1,  10^=1000,  100^=1000000,  and  so  on,  the 
cube  of  any  number  between  1  and  10  is  a  number  between 
1  and  1000,  the  cube  of  any  number  between  10  and  100  is 
a  number  between  1000  and  1000000,  and  so  on.  Thus  we 
see  that  the  cube  of  a  number  contains  three  times  as  many 
figures  as  the  number,  or  three  times  as  many  less  one-  or 
two.  If,  therefore,  a  number  be  separated  into  periods  of 
three  figures  each  by  placing  a  dot  over  every  third  figure, 
beginning  with  the  units'  figure,  the  number  of  figures  in 
the  root  equals  the  number  of  periods. 

Note  1.   The  left-hand  period  may  contain  but  one  or  two  figures. 

Note  2.  The  principle  applies  also  to  decimals,  because  the  cube  of 
a  decimal  contains  three  times  as  many  decimal  places  as  the  decimal 
itself. 

156.  The  component  parts  of  the  cube  of  a  number  of  two 
figures  may  be  learned  from  the  following  multiplication  : 

472=     2209=  402  +  2  X  (40  x7)  +  72 

47= 40  +  7 

15463  =  -  402  X  7  +  2  X  (40  X  72)  +  73 

8836    =  40"  +  2x  (40^x7)+  40  x  7^ 

47»  =  103823  =  40»  +  3  X  (40^  x  7)  +  3  x  (40  x  V)  +  7» 


INVOLUTION  AND  EVOLUTION. 


279 


In  general,  the  cube  of  any  number  composed  of  tens  and 
units  is  equal  to  the  cube  of  the  tens,  jflus  three  times  the 
product  of  the  square  of  the  tens  by  the  units,  plus  three  times 
the  product  of  the  tens  by  the  square  of  the  units,  plus  the  cube 
of  the  units. 

I.   Find  the  cube  root  of  175616. 

rj(j  Since    the    number 

consists  of  two  pe- 
riods, the  cube  root  will 
consist  of  two  figures. 
The  cube  of  the  tens' 
figure  of  the  root  must 
be  the  largest  cube  in 
175  thousands,  which 
is  125  thousands;  hence 
the  tens'  figure  of  the 
root  is  5.  Subtracting 
125000,  the  cube  of  60, 
from  175616,  the  re- 
mainder is  60016. 
Since  the  cube  of  the 
tens  has  been  subtract- 
ed, 6061^  equals  three  times  the  product  of  the  square  of  the  tens  by  the 
units,  p'ms  three  times  the  product  of  the  tens  by  the  square  of  the  units, 
plus  the  cube  of  the  units ;  this  is  the  same  as  three  times  the  square  of 
the  tens  plus  three  times  the  product  of  the  tens  by  the  units  plus  the 
square  of  the  units  multiplied  by  the  units.  Thus  the  two  factors  of 
50610  are  three  times  the  square  of  the  tens  plus  three  times  the 
product  of  the  tens  by  the  units  plus  the  square  of  the  units  and  the 
units.  We  can  find  the  units'  figure  by  dividing  50616  by  the  other 
factor ;  however,  this  factor  is  known  only  in  part,  so  we  take  three 
times  the  square  of  the  tens,  the  part  known,  for  the  trial  divisor,  and 
we  find  that  7500  is  contained  in  50616  six  times.  The  units'  figure, 
therefore,  is  6,  and  the  complete  divisor  is  7500  -1-  3  X  60  X  6  -f-  6^,  or 
8436.  Multiplying  8436  by  6,  we  obtain  50616,  and  there  is  no  further 
remainder.     Hence  the  cube  root  of  175616  is  56. 

If  the  number  consists  of  more  than  two  periods,  after 
finding  the  first  two  figures  of  the  root,  we  can  consider 


175616(50 -f  6 
125000 

3x502  =  7500 

3  X  50  X  G  =    900 

6^=     3G 

50616 

Or 

8436 

50616 

more  briefly 

17561( 
125 

3(56 

3 

7500 

900 

36 

5061( 

8436 

5061( 

5 

280 


ARITHMETIC. 


them  as  tens  in  reference  to  the  next  figure^  and  then  pro- 
ceed as  before. 

II.   Find  the  cube  root  of  22069.810125. 
22069.810125(28.05 


After  finding  28  in  the  root,  the 
trial  divisor  is  3x2802,  or  235200. 
This  is  not  contained  in  117810,  so 
the  next  figure  of  the  root  is  0.  Then 
the  trial  divisor  is  23520000,  which 
can  be  used  at  once  by  bringing  down 
another  period. 


The  quotient  obtained  by  dividing  the  dividend  by  the 
trial  divisor  often  proves  to  be  too  large ;  in  such  a  case  try 
a  smaller  figure  in  the  root. 

When  there  is  a  remainder  after  using  all  the  periods, 
further  figures  in  the  root  can  be  obtained  by  continuing 
the  process,  bringing  down  periods  of  three  zeros  each. 

The  cube  root  of  a  fraction  in  its  lowest  terms  may  be 
obtained  by  taking  the  cube  root  of  both  terms  when  they 
are  perfect  cubes.  When  either  term  is  not  a  perfect  cube, 
reduce  the  fraction  to  a  decimal  and  then  extract  the  root. 


1200 

14069 

480 

64 

1744 

13952 

23520000 

117810125 

42000 

25 

23562025 

117810125 

EXAMPLES. 

Find  the  cube  root  (to  four  decimal  places  when  the  num- 
ber is  not  a  perfect  cube)  of 


1.  4913. 

2.  24389. 

3.  250047. 

4.  636056000. 


5.  96702.579. 

6.  8365.427. 

7.  0.000032768. 

8.  0.001295029. 


9.  0.000000148877. 

10.  225866529. 

11.  1027243.729. 

12.  12000.8121619. 


INVOLUTION    AND   EVOLUTION.  281 

25.  f 

26.  tt. 

27.  15f. 

28.  12f 

29.  46f. 

30.  81,^. 


l.t 

0.27. 

19. 

0.61. 

14. 

10. 

20. 

51. 

15. 

1.025. 

21. 

1729. 

16. 

3.7. 

22. 

9358. 

17. 

0.0093. 

23. 

r%- 

18. 

18.65. 

24. 

,7_2_9 
1728* 

31.  Find  the  Vcilue  of  \(7.^j-^  to  three  decimal  places. 

\U.li6U 

32.  Find  the  value  of  yjz^  '^Tr^^  *^^®®  decimal  places. 

33.  Find  the  value  of  -v^  (1.05  V  to  three  decimal  places. 

34.  Find  the  value  of  y5  4-2\/6  to  two  decimal  places. 

35.  Multiply  2.49  by  22.32,  and  extract  ^he  cube  root  of 
the  product  to  two  decimal  places. 

36.  Divide  3.15  by  0.075,  and  extract  the  cube  root  of 

the  quotient  to  two  decimal  places. 

37.  Divide  6  by  0.89,  and  extract  the  cube  root  of  the 
quotient  to  two  decimal  places. 

38.  Multiply  108  billionths  by  two  thousand,  and  extract 
the  cube  root  of  the  product. 

39.  Multiply  -y/i  by  ^/0.456,  and  carry  the  result  to  two 
decimal  places. 

40.  Find  the  difference  between  the  sum  of  the  cube  roots 
of  32768  and  0.000512  and  the  cube  root  of  their  sum. 

41.  A  cubical  vessel  contains  1331' ;  what  is  ^he  lenjth 
of  its  edge  in  meters  ? 

42.  A  cubical  block  contains  12695.24  ca.  in. ;   fiiid  th^ 

lenath  of  one  side. 


282 


ARITHMETIC. 


/'X4 

/ 

/ 

d      9 

C 

/ 

h      / 

Fia.  1. 


Fig.  2. 


Fig. 


13824(24 
8 


157.   Cube  root  can  also  be  explained  by  the  aid  of  dia- 
grams. 

The  solid  contents  of  a  cube  are  found  by  cubing  the  length  of  an 
edge;  hence  the  length  of  an  edge  may  be  found  by  extracting  the 

cube  root  of  the  number  denoting 
the  solid  contents. 

Let  Fig.  1  represent  a  cube  con- 
taining 13824  cubic  units ;  we  wish 
to  determine  the  length  of  an  edge. 

Since    the   number  denoting   the 

cubic   contents   consists  of   two  pe- 

//y/i       riods,  the  cube  root  will   consist  of 

y  a    X  h  X  '^yYcrr?       ^^^  figures.     The  cube  of  the  tens' 

^ figure  of  the  root  must  be  the  largest 

cube  in  13  thousands,  which  is  8 
thousands ;  hence  the  tens'  figure  of 
the  root  is  2,  and  the  length  of  an 
edge  of  the  cube  is  20  plus  the  units' 
figure  of  the  root.  Let  Fig.  2  repre- 
sent a  cube  whose  edge  is  20  units ; 
then  its  volume  is  8000.    Subtracting 

^ this   from    13824,  we   have   left   an 

irregular  solid  whose  volume  is  5824. 
This  ^rfegular  solid  consists  of  the  three  rectangular  solids  a,  h,  and 
c,  th*.  three  smaller  rectangular  solids  d,  e,  and  f,  and  the  cube  g. 
Arranging  them  as  in  Fig.  3,  we  have  a  series  of  solids  which  have  for 
their  common  thickness  the  units'  figure  of  the  root.  The  bases  of  a,  6, 
and  c  are  each  20^,  or  400;  the  bases  of  d,  e,  and/ are  each  tlie  product 
of  20  and  the  units'  figure  of  the  root ;  and  the  base  of  g  is  the  square  of 
the  units'  figure  of  the  root.  When  the  volume  and  base  of  a  rectangu- 
lar solid  are  known,  the  thickness  can  be  found  by  dividing  the  volume 
by  the  base ;  the  common  thickness  of  the  solids  in  Fig.  3  can  be  found 
by  dividing  their  entire  volume  by  the  sum  of  their  bases,  a,  b,  and  c 
are  the  only  solids  whose  bases  are  known,  so  we  take  the  sum  of  these 
bases,  3  X  20^,  or  1200,  for  the  trial  divisor,  and  by  dividing  5824  by 
1200,  we  have  4  for  the  units'  figure  of  the  root.  The  sum  of  the  bases, 
then,  is  3  X  202  +  3  X  20  X  4  -f  42,  or  1456.  The  product  of  1456  and  4  is 
5824,  and  there  is  no  further  remainder.  Hence  24  is  the  cube  root  of 
13824. 


1200 

240 
16 


1456 


5824 


5824 


IKVOLUTIOK  AND   EVOLUTION.  288 

Higher  Roots. 

158.  By  means  of  the  processes  of  square  and  cube  root, 

we  can  find  any  root  whose  index  contains  only  the  factors 
2  and  3.  For  example,,  the  fourth  root  of  a  number  may  be 
found  by  taking  the  square  root  of  the  square  root;  the 
sixth  root  of  a  number  may  be  found  by  taking  the  square 
root  of  the  cube  root  or  the  cube  root  of  the  square  root. 

EXAMPLES. 

1.  Find  the  fourth  root  of  1874161. 

2.  Find  the  fourth  root  of  8.25  to  tliree  decimal  places. 

3.  Find  the  sixth  root  of  1291467969. 

4.  Find  the  sixth  root  of  0.184  to  three  decimal  places. 

5.  Find  the  eighth  root  of  2562890625. 

6.  Find  the  eighth  root  of  5  to  three  decimal  places. 

7.  Find  the  ninth  root  of  134217728. 

8.  Find  the  ninth  root  of  3^  to  two  decimal  places. 

9.  Find  the  twelfth  root  of  13841287201. 

10.   Find  the  twelfth  root  of  0.75  to  two  decimal  places. 


284  ARITHMETIC. 

CHAPTER  XIII. 
SERIES. 

159.  A  series  is  a  succession  of  numbers,  each  of  which 
is  derived  from  one  or  more  of  the  preceding  by  a  fixed 
law.  The  numbers  which  compose  a  series  are  called  its 
terms ;  the  first  and  last  terms  are  called  the  extremes,  and 
the  other  terms  are  called  the  means.  An  ascending  series 
is  one  in  which  the  terms  increase  regularly  from  the  first 
term  ;  a  descending  series  is  one  in  which  the  terms  decrease 
regularly  from  the  first  term. 

Arithmetical  Progression. 

160.  When  a  series  increases  or  decreases  by  a  common 
difference,  it  is  called  an  arithmetical  series  or  arithmetical 
progression.  For  example,  2,  5,  8,  11,  14,  17  is  an  ascend- 
ing arithmetical  progression,  in  which  the  common  differ- 
ence is  3. 

In  every  arithmetical  progression  there  are  five  elements 
to  be  considered,  —  the  first  term,  the  last  term,  the  common 
difference,  the  number  of  terms,  and  the  sum  of  the  terms. 
These  five  elements  bear  such  a  relation  to  each  other  that 
when  any  three  are  given,  the  other  two  can  be  found. 
This  gives  rise  to  twenty  distinct  cases,  a  few  of  the  more 
important  of  which  will  here  be  illustrated. 

I.  In  an  ascending  arithmetical  series  the  first  term  is  7, 
and  the  common  difference  is  4 ;  find  the  10th  term. 

_       „        .  „  „        .  „  The  first  term  is  7 ;  the  second  term 

■"  "'  '       equals  7  plus  the  common  difference  ; 

the  third  term  equals  the  second  term  plus  the  common  difference,  or 


SERIES.  285 

7  plus  twice  the  common  difference ;  the  fourth  term  equals  the  third 
term  plus  the  common  difference,  or  7  plus  three  times  the  common 
difference.  In  like  manner,  the  tenth  term  equals  7  plus  nine  times 
the  common  difference. 

II.  Find  the  first  term  of  an  ascending  arithmetical 
series,  the  last  term  of  which  is  47,  the  common  difference 
6,  and  the  number  of  terms  8. 

._       _       n  _  Arr       A(y r  '^^^^  ^^^^  \,^xxQ.  Hiust  bc  such  a  num- 

47  —  ^  X  0  —  4/  —  4Z  _  o.    ^^^  ^^^^  .^  ^  ^g  ^g  ^j^^^  ^^  .^^  ^^^ 

result  will  be  47 ;  hence,  if  7  X  6  be  subtracted  from  47,  the  remainder 
is  the  first  term. 

III.  The  extremes  of  an  arithmetical  progression  are  8 
and  63,  and  the  number  of  terms  is  12 ;  what  is  the  com- 
mon difference  ? 

The  last  term  is  determined  by  add- 
63  —  8  _  ^  __  5  ing  the  common  difference  to  the  first 

12  —  1       11         *  term  as  many  times  as  there  are  terms 

less  one;  hence  63  —  8,  the  difference 
between  the  extremes,  equals  the  common  difference  taken  12—1  times, 
and  the  common  difference  equals  63  —  8  divided  by  12  —  1. 

IV.  The  extremes  of  an  arithmetical  progression  are  4 
and  103,  and  the  common  difference  is  9 ;  what  is  the  num- 
ber of  terms  ? 

^  ^f^       .  The  difference  between  the  extremes 

(-  1  =  11  -j-1  =  12.    equals  the  common  difference  taken  as 

"  many  times  as  there  are  terms  less  one ; 

hence  J-^f=^  equals  one  less  than  the  number  of  terms,  and  J-^|^^+  1 
equals  the  "number  of  terms. 

V.  The  first   term  of  an  arithmetical  progression  is  4, 

the  last  term  19,  and  the  number  of  terms  6 ;  find  the  sum 

of  the  terms. 

A  V  /'/i  _L1  Q\       A  v/  OQ  Using  the  method  shown  in  example 

o  X  y^-^rd)  _  D  X  ^c»  _  go  19_4 

9  9  ■  III.,  the  common  difference  is  , 

which  equals  V,  or  3.     The  sum  of  the  series  can  then  be  writteo 


286  ARITHMETIC. 

4+    7+10  +  13  +  16  +  19, 
or  19  +  16  +  13  +  ID  +    7  +    4.     By  addition  we  find 
twice  the  sum  to  be    23  +  23  +  23  +  23  +  23  +  23, 
which  equals  6  X  (4  +  19)  ;  hence  the  sum  of  the  terms  equals 
6x(4+19) 
2 

Let  the  first  term  be  represented  by  a,  the  last  term  by  ?, 
the  common  difference  by  d,  the  number  of  terms  by  n,  and 
the  sum  of  the  terms  by  s ;  then  the  principles  illustrated  in 
the  foregoing  examples  maybe  briefly  expressed  as  follows ; 


1= 

■■a  +  {n-l)x 

d. 

a- 

:  l-{n-l)x 

d. 

d  = 

l-a 
n-1 

n  = 

d 

.= 

:|x(a  +  0. 

Note.  The  principles  as  stated  apply  to  ascending  series.  They  can 
be  stated  so  as  to  apply  to  descending  series  by  interchanging  a  and  /. 

To  solve  a  problem  in  arithmetical  progression,  it  is 
merely  necessary  to  substitute  the  given  values  in  the 
proper  formula,  and  then  simplify  the  expression  thus  ob' 
tained. 

EXAMPLES. 

1.  The  first  term  of  an  ascending  arithmetical  series  is 
6,  and  the  common  difference  is  5 ;  find  the  20th  term. 

2.  In  an  ascending  arithmetical  series  the  first  term  is 
8,  and  the  common  difference  is  f ;  what  is  the  30th  term  ? 

3.  The  first  term  of  a  descending  arithmetical  series  is 
120,  the  common  difference  6,  and  the  number  of  terms  15 ; 
what  is  the  last  term  ? 


SBEIES.  287 

4.  The  first  term  of  a  descending  arithmetical  series  is 
64,  and  the  common  difference  is  4 ;  find  the  12th  term. 

5.  The  12th  term  of  an  ascending  arithmetical  series  is 
60,  and  the  common  difference  is  2 ;  what  is  the  first  term  ? 

6.  The  last  term  of  a  descending  arithmetical  series  is 
3,  the  common  difference  4,  and  the  number  of  terms  11 ; 
what  is  the  first  term  ? 

7.  The  extremes  of  an  arithmetical  progression  are  9 
and  49,  and  the  number  of  terms  is  9 ;  what  is  the  common 
difference  ? 

8.  The  extremes  of  an  arithmetical  progression  are  l^- 
and  24,  and  the  common  difference  is  2^ ;  find  the  number 
of  terms. 

9.  The  extremes  of  an  arithmetical  progression  are  0 
and  150,  and  the  number  of  terms  is  16 ;  find  the  sum  of 
all  the  terms. 

10.  Find  the  sum  of  the  first  12  terms  of  the  series  3,  7, 
11,  etc. 

11.  Find  the  sum  of  the  first  10  terms  of  the  series  24, 
22i,  21,  etc. 

12.  Find  the  sum  of  the  odd  numbers  from  1  to  49  in- 
clusive. 

13.  How  many  strokes  does  the  hammer  of  a  clock  strike 
in  12  hours  ? 

14.  How  far  can  a  man  walk  in  10  days,  going  12  miles 
the  first  day,  and  increasing  the  rate  3  miles  a  day  ? 

15.  A  man  travelled  13  days,  travelling  each  day  |  of  a 
mile  more  than  the  preceding  day.  If  he  went  18  miles 
the  last  day,  how  many  miles  did  he  travel  the  first  day  ? 

16.  A  man  going  a  journey,  travelled  the  first  day  5 
miles,  the  last  day  32  miles,  and  each  day  3  miles  more  than 
the  preceding  day.     How  many  days  did  he  travel  ? 


288  ARITHMETIC. 

17.  A  man  travels  11  days,  travelling  5  miles  the  first 
day,  and  increasing  the  distance  equally  each  day,  so  that 
the  last  day's  journey  is  20  miles  ;  find  the  daily  increase. 

18.  A  man  has  8  children,  whose  several  ages  differ 
alike;  the  youngest  is  2  years  old,  and  the  oldest  30. 
What  is  the  common  difference  of  their  ages  ? 

19.  A  laborer  worked  for  40  cents  the  first  day,  and  on 
each  succeeding  day  his  wages  were  increased  5  cents  ;  on 
the  last  day  he  received  $2.50.  How  many  days  did  he 
work? 

20.  A  stone  falls  16.08  ft.  during  the  first  second,  48.24 
ft.  during  the  next  second,  80.4  ft.  during  the  third  second, 
and  so  on ;  how  far  will  it  fall  during  the  ninth  second  ? 
How  far  will  it  fall  in  nine  seconds  ? 

21.  Find  the  sum  of  an  arithmetical  series  whose  first 
two  terms  are  6  and  13,  and  whose  last  term  is  62. 

22.  Find  the  amount  of  $300  for  15  years  at  5%  simple 
interest. 

23.  In  how  many  years  will  $150  amount  to  $330  at  6% 
simple  interest  ? 

24.  The  amount  of  $500  for  18  years  is  $860 ;  what  is 
the  yearly  interest  ? 

Geometrical  Progression. 

161.  When  a  series  increases  or  decreases  by  a  common 
ratio,  it  is  called  a  geometrical  series  or  geometrical  pro- 
gression. For  example,  2,  6,  18,  54,  162  is  an  ascending 
geometrical  progression,  in  which  the  ratio  is  3. 

An  infinite  series  is  a  descending  series  of  an  infinite 
number  of  terms.  For  example,  1,  J,  |,  ^,  -^L  etc. ;  the  last 
term  is  infinitely  small  and  is  regarded  as  zero, 


SERIES.  289 

In  every  geometrical  progression  there  are  five  elements 
to  be  considered, —^Ae^rs^  term,  the  last  term^  the  ratio,  the 
number  of  terms,  and  the  sum  of  the  terms.  As  in  arithmeti- 
cal progression,  the  five  elements  bear  such  a  relation  to 
each  other  that  when  any  three  are  given,  the  other  two 
can  be  found. 

I.  The  first  term  of  a  geometrical  series  is  2,  and  the 
ratio  is  3 ;  what  is  the  7th  term  ? 

2  X  3^^  =  2  X  3«  =  2  X  729  =  1458.        '^^^  ^"*  *"'"^  ,"  f '  *^," 

second  term  equals  2  mul- 
tiplied by  the  ratio;  the  third  term  equals  the  second  term  multiplied 
by  the  ratio,  or  2  multiplied  by  the  square  of  the  ratio;  the  fourth 
term  equals  the  third  term  multiplied  by  the  ratio,  or  2  multiplied  by 
the  cube  of  the  ratio.  In  like  manner,  the  seventh  term  equals  2  mul- 
tiplied by  the  sixth  power  of  the  ratio. 

II.  The  last  term  of  a  geometrical  series  is  640,  the  ratio 
2,  and  the  number  of  terms  8 ;  what  is  the  first  term  ? 

640      640      640  ^^®  ^^^^  *^'™  °^"^*  ^'^  *"^^  *  number 

98^  =  ~-^  =  r^  =  ^«  that  if  it  be  multiplied  by  2^,  the  result  will 

^        ^"^^  be  640;  hence,  if  040  be  divided  by  2^  the 
quotient  is  the  first  term. 

III.  The  first  term  of  a  geometrical  series  is  7,  the  last 
term  567,  and  the  number  of  terms  5 ;  what  is  the  ratio  ? 

5-1 1567  S67,  the  fifth  term,  equals  7  multiplied  by 

"~=  V81  =  3.  the  4th  power  of  the  ratio ;  hence,  if  567 

be   divided   by  7,  the  quotient,  81,  is  the 
fourth  power  of  the  ratio,  and  the  ratio  is  the  4th  root  of  81,  or  3. 

IV.  The  extremes  of  a  geometrical  progression  are  8  and 
768,  and  the  ratio  is  4 ;  find  the  sum  of  the  terms. 

768  X  4 - 3 _ 3072-3 _ 3069     ^ ^^q 
4-1       -        3^  - ^- -=^^^^- 

The  sum  of  the  series  may  be  written 


290  ARITHMETIC. 

3  +  12  +  48  +  192  +  768.  Four  times  the  sura 

equals  12  +  48  +  192  +  768  +  3072.  Subtracting 

the  upper  line  from  the  lower  line,  we  have  3072  — 3,  which  equals 
three  times  the  sum  of  the  series ;  hence  the  sum  of  the  series  equals 


or  1023. 


3 

Note.   The  last  term,  768,  equals  the  first  term  multiplied  by  4^-1 ; 
hence  768  X  4,  or  3072,  equals  the  first  term  multiplied  by  4^,  and  the 

value  of  the  sum  might  be  written  — — ^^^,  which  equals  1023. 

4  —  1 

V.   Find  the  sum  of  the  first  5  terms  of  the  series  768, 

192,  48,  etc. 


768  -3  X  i:^768  -  |^  767^^3069 

1   _  1  3  8  Q 


1023. 


The  sum  of  the  series  may  be  written 

768  +  192  +  48+12  +  3.     One  fourth  of  the  sum 
equals  192  +  48  +  12  +  3  +  f .  Subtracting 

the  lower  line  from  the  upper  line,  we  have  768  —  |,  which  equals  three 
fourths  of  the  sum  of  the  series ;  hence  the  sum  of  the  series  equals 

^^^^,  or  1023. 

3  * 

Note.  The  value  of  the  sum  might  be  written  '^Q  — 768x(^)  ^^^^i^.]^ 
equals  1023.  ^~? 

VI.   rind  the  sum  of  the  infinite  series  768,  192,  48,  etc. 

768    _  768  _  3072  _  ..  ^^^a  The  last  term  is  regarded  as  0 ; 

1  _  1  3  3  hence  the  expression  for  the  sum  as 

found  in  the  preceding  example  be- 
comes 768-0 X^  ^jjj^jjj  g       ig  768  ^^  JQ24. 
1-i  f 

Let  the  first  term  be  represented  by  a,  the  last  term  by  I, 
the  ratio  by  r,  the  number  of  terms  by  n,  and  the  sum  of 
the  terms  by  s ;  then  the  principles  illustrated  in  the  fore- 
going examples  may  be  briefly  expressed  as  follows : 

I 
a  —  — -• 


SERIES.  291 


T  \C  f  —  d  Qi  ^  9*"  -—  CI 

s  =  -^ or  J    for  ascending  series. 

r  —  1  7'  —  1 

a  —  lxr        a  —  axr^'o      -,  ■,. 

8  = or   >    tor  descending  series. 

1 — r  1—r 

for  infinite  series. 


1-r 

To  solve  a  problem  in  geometrical  progression,  it  is  merely 
necessary  to  substitute  the  given  values  in  the  proper  for- 
mula, and  then  simplify  the  expression  thus  obtained. 

EXAMPLES. 

1.  The  first  term  of  a  geometrical  series  is  8,  and  the 
ratio  is  4 ;  find  the  8th  term. 

2.  The  first  term  of  a  geometrical  series  is  27,  and  the 
ratio  is  ^ ;  find  the  7th  term. 

3.  The  6th  term  of  a  geometrical  series  is  3888,  and  the 
ratio  is  6 ;  find  the  first  term. 

4.  The  last  term  of  a  geometrical  series  is  60f ,  the  ratio 
£,  and  the  number  of  terms  7 ;  what  is  the  first  term  ? 

5.  The  first  term  of  a  geometrical  series  is  10,  and  the 
6th  term  is  2430 ;  what  is  the  ratio  ? 

6.  The  first  term  of  a  geometrical  series  is  y^,  the  last 
term  104J,  and  the  number  of  terms  7 ;  find  the  ratio. 

7.  Find  the  sum  of  the  first  9  terms  of  the  series  whose 
first  term  is  6  and  ratio  4. 

8.  Find  the  sum  of  the  first  5  terms  of  the  series  whose 
first  term  is  100  and  ratio  I. 

9.  Find  the  sum  of  an  infinite  series  whose  first  term  is 
3  and  ratio  J. 


292  ARITHMETIC. 

10.  Find  the  sum  of  the  first  8  terms  of  the  series  8,  4,  2, 
etc.     Find  the  sum  of  the  same  series  to  infinity. 

11.  The  second  term  of  a  geometrical  progression  is  36 ; 
find  the  sum  of  4  terms  when  the  ratio  is  1 J ;  also  when  the 
ratio  is  1|. 

12.  A  merchant  doubles  his  capital  every  5  years ;  if  he 
begins  with  $2000,  how  much  has  he  at  the  end  of  25  years  ? 

13.  If  a  ball  be  put  in  motion  by  a  force  which  would 
move  it  10  feet  the  first  second.  8  feet  the  second,  6.4  feet 
the  third,  and  so  on,  how  far  would  it  move  ? 

14.  What  sum  of  money  can  be  paid  by  10  instalments, 
the  first  of  which  is  $1,  the  second  $2,  the  third  f  4,  and 
so  on  in  a  geometrical  progression  ? 

15.  A  man  worked  8  days  on  condition  that  he  should 
receive  1  cent  the  first  day,  5  cents  the  second  day,  and  so 
on,  the  wages  of  each  day  being  5  times  the  wages  of  the 
previous  day ;  how  much  did  he  receive  ? 

16.  A  man  travels  4  miles  the  first  day,  8  miles  the  sec- 
ond day,  16  miles  the  third  day,  and  so  on.  How  far  does 
he  travel  the  7th  day  ?     How  far  does  he  travel  in  7  days  ? 

Compound  Interest. 

162.  Problems  in  compound  interest  can  be  solved  by 
the  principles  of  geometrical  progression. 

Let  P  represent  the  principal,  r  the  interest  of  fl  for  1 
year,  n  the  number  of  years,  and  A  the  amount  of  the  given 
principal  for  n  years.  In  computing  compound  interest  / 
is  multiplied  by  1+^'  as  many  times  as  there  are  years. 
Thus  P  is  the  first  term  of  a  geometrical  series,  of  which  A 
is  the  last  term,  and  l-\-r  the  ratio ;  the  number  of  terms  ia" 


SERIES. 


203 


one  more  than  the  number  of  years,  or  n+1.  Hence  the 
first  three  formulae  for  geometrical  progression  when  applied 
to  compound  interest  become 

A  =  Px(l+r)\ 

A 


(l+ry 


l  +  r  = 


Sp 


or  r 


-<li 


I.   Find  the  amount  of  f  250  for  3  years  at  5%  compoauu 
interest. 


To  solve  the  problem  it  i» 
merely  necessary  to  substitute 
the  values  for  P,  r,  and  n  in 
the  formula  for  A,  and  then 
simplify. 


^  =  250x  (1.05)3. 

1.05 
1.05 
525 
105 

1.157625 
250 

57881250 
2315250 
289.406250 

1.1025 
1.05 

Ans.  $289.41 

55125 

11025 

1.157625 


II.   At  what  rate  per  cent  must  $500  be  put  out  at  com- 
pound interest  to  amount  to  $571.20  in  3  years  ? 


=^ 


571.20      . 

500)571.20 

^'      "^        ^   *  Substitute  values  of  A,  P,  and  n  in  the 

formula  for  r,  and  then  simplify. 


30000 
1200 

16 

31216 


142400 


124864 


Ans.  4.5%. 


3244800)17536000 


294  ARITHMETIC. 


EXAMPLES. 


1.  Find  the  amount  of  $600  for  5  years  at  6%  com- 
pound interest. 

2.  Find  the  amount  of  $1500  for  6  years  at  7%  com- 
pound interest. 

3.  Find  the  amount  of  $75  for  8  years  at  4%  compound 
interest. 

4.  What  principal  at  5%  compound  interest  will  amount 
^0  153.98  in  4  years  ? 

5.  What  principal  at  6%  compound  interest  will  amount 
to  f  1000  in  6  years  ? 

6.  What  sum  must  be  invested  at  4%  compound  inter- 
est to  amount  to  $600  in  8  years  ? 

T .  At  what  rate  of  compound  interest  will  $2500  amount 
in  3  years  to  $4320  ? 

8.  At  what  rate  of  compound  interest  will  $500  amount 
in  4  years  to  $631.24  ? 

9.  At  what   rate  of  compound  interest  will  a  sum  of 
money  double  itself  in  6  years  ? 

10.  At  what  rate  of  compound  interest  will  a  sum  of 
money  treble  itself  in  12  years  ? 

Annuities. 

163.  An  annuity  is  a  sum  of  money,  payable  yearly,  to 
continue  for  a  certain  number  of  years,  for  life,  or  forever. 
The  term  is  also  applied  to  a  sum  of  money  payable  at  any 
regular  intervals  of  time. 

The  amount  or  final  value  of  an  annuity  is  the  sum  of  all 
the  payments  plus  the  interest  on  each  payment  from  the 
time  it  becomes  due  until  the  annuity  ceases. 


SERiES.  295 

The  present  worth  of  an  annuity  is  snch  a  sum  of  money 
as  will^  if  put  at  interest,  amount  to  the  final  value. 

Annuities  at  Simple  Interest. 

164.  Problems  in  annuities   at  simple  interest  can  be 
solved  by  the  principles  of  arithmetical  progression. 

I.  What  is  the  amount  of  an  annuity  of  $300  for  5  years 
at  6%  simple  interest? 

1  =  a  +  {n-l)xd         «_Wv/'^a.7^  The  payment  at 

0/.A   .  /r      ^^      .Q   *~o  ^v"'^*>'  the  end  of  the  fifth 

=  300-H(5-l)xl8  j  year  is  ^00,  the 

=  300  +  72  =  372.  =  ^  X  (300  +  372)     payment  at  the  end 

of  the  fourth  year 
^^iin  «iAcn  amounts  to  $300 
^  plus    the    mterest 

for  1  year,  the  pay- 
ment at  the  end  of  the  third  year  amounts  to  $300  plus  the  interest  for 

2  years,  and  so  on.  These  sums  form  an  arithmetical  progression,  of 
which  $300  is  the  first  term,  the  interest  of  $300  for  1  year,  or  $18,  is 
the  common  difference,  and  5  is  the  number  of  terms.     By  principles 

roi  arithmetical  progression,  we  find  the  sum  of  the  terms  to  be  $1680. 

To  find  the  present  worth  of  the  annuity  of  example  I., 
find  the  present  worth  of  $1680  for  5  years,  as  shown  in  §  138. 

II.  Find  the  annuity  whose  amount  for  6  years  at  5% 
simple  interest  is  $1350. 

lz=a-{-{n  —  l)xd 

=  1 +(6 -l)x. 05=1.25.      ^.  ,,  .,  ^, 

'  \  ^  If  the  annuity  were  $1,  we  Wb-uld 

s  =  -(a+l)  have a=  1,  /=  1  +(6-l)x.06=  1.25, 

2  and  s  =  f  (1  +  1.26)  =  6.75.     It  takes 

6  x^  _,   ^  nK\ a  TK  *^  annuity  of  as   many  dollars  to 

"2'  ^  ^^  amount  to  $1350  as  $6.75  is  con- 

6  TS'ilS^O  00<'$200  tained  times  in  $1350,  which  equals 


1350_ 
00 


$200. 


296  ARITHMETIC. 


EXAMPLES. 

1.  What  is  tlie  amount  of  an  annuity  of  $500  for  8 
years  at  6%  simple  interest? 

2.  What  is  the  amount  of  an  annuity  of  $1000  for  10 
years  at  5%  simple  interest  ? 

3.  What  is  the  amount  of  an  annuity  of  $200  for  5 
years  at  4|%  simple  interest? 

4.  A  clerk's  salary  of  $1000  a  year,  payable  quarterly, 
remained  unpaid  for  three  years  ;  find  the  amount  then  due, 
reckoning  interest  at  6%. 

5.  What  is  the  present  worth  of  an  annuity  of  $600  for 
6  years  at  6%  simple  interest  ? 

6.  What  is  the  present  worth  of  an  annuity  of  $150  for 
12  years  at  4%  simple  interest  ? 

7.  Find  the  annuity  whose  amount  for  6  years  at  6% 
simple  interest  is  $3450. 

8.  Pind  the  annuity  whose  amount  for  10  years  at  5% 
simple  interest  is  $3675. 

9.  Find  the  annuity  whose  amount  for  5  years  at  4% 
simple  interest  is  $1500. 

Annuities  at  Compound  Interest. 

165.  Problems  in  annuities  at  compound  interest  can  be 
solved  by  the  principles  of  geometrical  progression. 

I.  What  is  the  amount  of  an  annuity  of  $200  for  4  years 
at  6%  compound  interest  ? 

^  g  X  r"  -  g  ^  200  X  (1.06)^-  200 
^         r~l  1.06-1 


SERIES. 


297 


The  payment  at  the  end  of 
the  fourth  year  is  $'200,  the 
payment  at  the  end  of  the 
third  year  amounts  to  $200 
plus  the  interest  for  1  year, 
the  payment  at  the  end  of  the 
second  year  amounts  to  $200 
plus  the  compound  interest  for 
2  years,  and  so  on.  These 
sums  form  a  geometrical  pro- 
gression, of  which  $200  is  the 
first  term,  1.00  is  the  ratio, 
and  4  is  the  number  of  terms. 
By  the  principles  of  geometrical 
progression,  we  find  the  sum  of 
the  terms  to  be  $874.92. 

To  find  the  present  worth  of  the  annuity  of  example  I., 
find  the  sum  of  money  which  put  on  interest  for  4  years  at 
6%  compound  interest  will  amount  to  f  874.92,  as  shown  in 
§  162. 

II.  Find  the  annuity  whose  amount  for  3  years  at  5^^ 
compound  interest  is  f  504.40. 

^axr-a^l  x(1.05)»-l 
^         r-1  1.05-1 


1.06 
1.06 

1.26247696 

200 

636 
106 

252.49539200 
200 

1.1236 
1.06 
67416 
11236 

.06)52.495392 

^874.92 

1.191016 
1.06 

7146096 
1191016 

1.26247696 

55125 
11025 

1.157625 

L 

.05). 157625 
3.1525 


1.05 
1.05 

3.1525)504.4000(1160 
31525 

525 

189150 

105 

189150 

1.1025 

0 

1.05 

If  the  annuity  were  $1, 
we  would  have  a=l,  and 
^^1X005)^^3,52, 

1.05-1 
It  takes  an  annuity  of  as 
many  dollars  to  amount  to 
$504.40  as  $3.1525  is  con- 
tained times  in  $504.40, 
which  equals  f  _60. 


298  ARITHMETIC. 

EXAMPLES. 

1.  What  is  the  amount  of  an  annuity  of  $50  for  5  years 
at  6%  compound  interest  ? 

2.  What  is  the  amount  of  an  annuity  of  $200  for  6 
years  at  7%  compound  interest? 

3.  What  is  the  amount  of  an  annuity  of  $1000"  for  8 
years  at  4%  compound  interest  ? 

4.  If  a  man  deposits  f  100  a  year  in  a  savings  bank  that 
pays  3%  compound  interest,  how  much  will  he  have  in  the 
bank  at  the  end  of  10  years  ? 

5.  What  is  the  present  worth  of  an  annuity  of  f  60  for 
4  years  at  6%  compound  interest  ? 

6.  What  is  the  present  worth  of  an  annuity  of  $200  for 
7  years  at  5%  compound  interest  ? 

7.  Find  the  annuity  whose  amount  for  3  years  at  6% 
compound  interest  is  $95.51. 

8.  Find  the  annuity  whose  amount  for  5  years  at  6% 
compound  interest  is  $2818.55. 

9.  Find  the  annuity  whose  amount  for  6  years  at  5% 
compound  interest  is  $1000. 


MENSUKATION.  299 


CHAPTER  XIV. 

MENSURATION. 

106.  Mensuration  is  the  process  of  finding  the  lengths  of 
lines,  the  areas  of  surfaces,  or  the  volumes  of  solids.  The 
principles  of  mensuration  that  apply  to  rectangles  and  rec- 
tangular solids  are  given  in  sections  80  and  81.  The  present 
chapter  contains  such  principles  as  are  useful  to  students 
of  Arithmetic,  but  the  proofs  of  these  principles  must  be 
learned  from  Geometry. 

Definitions. 

167.   A  point  is  that  which  has  only  position. 

A  line  is  that  which  has  length  without  breadth  or  thick- 
ness. A  straight  line  is  a  line  which  has  the  same  direction 
throughout  its  wliole  extent.  A  curved  line  is  a  line  which 
changes  its  direction  at  every  point. 


Straight  Line.  Curved  Linx. 

A  surface  is  that  which  has  length  and  breadth  without 
thickness.  A  plane  surface  is  a  surface  such  that  if  any 
two  of  its  points  be  joined  by  a  straight  line,  that  line  lies 
wholly  in  the  surface.  A  curved  surface  is  a  surface  no 
portion  of  which  is  plane. 

A  solid  is  that  which  has  length,  breadth,  and  thickness. 

Parallel  lines  are  lines  in  the  same 
plane  which  have  the  same  direction.  _"" 

They  are  equally  distant  and  can  never  parallel  lineb. 

meet. 


300 


ARITHMETIC. 


Right  Aitgle. 


Obtusb  Angle. 


Acute  Angle. 


An  angle  is  the  difference  in  direc. 
tion  between  two  lines  which  meet  at 
a  point,  called  the  vertex;  the  lines 
are  called  the  sides  of  the  angle.  BAC 
is  an  angle  whose  vertex  is  the  point 
A,  and  whose  sides  are  AB  and  AC. 

A  right  angle  is  an  angle  such  that 
if  one  of  its  sides  be  produced  through 
the  vertex,  the  two  angles  thus  formed 
are  equal.  The  two  sides  of  a  right 
angle  are  said  to  be  perpendicular  to 
each  other. 

All  angles  not  right  angles  are  called 
oblique  angles.  An  angle  greater  than 
a  right  angle  is  called  an  obtuse  angle, 
and  an-  angle  less  than  a  right  angle  is 
called  an  acute  angle. 
A  plane  figure  is  a  portion  of  a  plane  surface  bounded  by 
straight  or  curved  lines. 

A  polygon  is  a  plane  figure  bounded  by  straight  lines ;  the 
lines  are  called  the  sides  of  the  polygon.  The  perimeter  of 
a  polygon  is  the  sum  of  its  sides.  The  area  of  a  polygon 
is  the  surface  included  within  the  perimeter.  A  diagonal 
of  a  polygon  is  a  line  joining  the  vertices  of  two  angles  not 
adjacent. 

An  equilateral  polygon  has  all  its  sides  equal.  An  equi- 
angular polygon  has  all  its  angles  equal.  A  regular  polygon 
is  both  equilateral  and  equiangular. 

Polygons  are  named  according  to  the  number  of  sides. 
A  polygon  of  three  sides  is  a  triangle,  four  sides  a  quadri- 
lateral, five  sides  a  pentagon,  six  sides  a  hexagon,  seven 
sides  a  heptagon,  eight  sides  an  octagon,  nine  sides  a  nona^ 
gon,  ten  sides  a  decagon,  and  so  on. 


MENSURATION. 


301 


Triangles. 

168.  A  polygon  of  three  sides  is  called  a  triangle.     The 

base  of  a  triangle  is  the  side  on  which  it  is  supposed  to 

stand.    The  vertical  angle  is  the  angle 

opposite  the  base,  and  its  vertex  is 

called  the  vertex  of  the  triangle.    The 

altitude  is  the  perpendicular  distance 

from  the  vertex  to  the  base.      The 

base  and  altitude  are  called  the  dimen- 
sions of  the  triangle.     In  the  triangle 

ABC,  BG  is  the  base,  the  angle  BAO  is  the  vertical  angle, 

and  AD  is  the  altitude. 
An  equilateral  triangle  has  all  its  sides  equal.     (Fig.  1.) 
An  isosceles  triangle  has  two  of  its  sides  equal.    (Fig.  2.) 
A  scalene  triangle  has  no  two  sides  equal.     (Fig.  3.) 


Pio.  1. 


Fig.  2. 


Fig.  3. 


Fig.  4. 


A  right  triangle  is  a  triangle  having  one  right  angle. 
The  side  opposite  the  right  angle  is  called  the  hypotenuse 
or  hypothenuse,  and  the  side  perpendicular  to  the  base  is 
called  the  perpendicular.  Fig.  4  is  a  right  triangle,  in 
which  AB  is  the  hypotenuse,  CB  the  base,  and  AC  the 
perpendicular. 

An  obtuse  triangle  is  a  triangle  having  one  obtuse  angle. 
(Fig.  3.) 

An  acute  triangle  is  a  triangle  having  three  acute  angles. 
(Fig.  2.) 

An  ec^niangalar  triangle  has  all  its  angles  eijual.  (Fig.  1.) 


302  ARITHMETIC. 

To  find  the  area  of  a  triangle  wlien  the  base  and  altitude 
are  given,  take  one  half  the  product  of  the  base  by  the  altitude. 

To  find  the  area  of  a  triangle  when  three  sides  are  given, 
subtract  each  side  separately  from  half  the  sum  of  the  three 
sides;  then  multiply  the  continued  product  of  these  three  re- 
mainders by  half  the  sum  of  the  sides,  and  extract  the  square 
root  of  the  product. 

To  find  one  dimension  of  a  triangle  when  tfte  area  and 
the  other  dimension  are  given,  divide  twice  the  area  by  the 
given  dimension. 

Let  A  represent  the  area  of  a  triangle,  h  the  altitude,  b 
the  base,  a  and  c  the  other  two  sides,  and  s  half  the  sum  of 
the  sides ;  then 

A^^Xbxh. 

A  =  Vs  X  {s  —  a)  X  {s  —  b)  X  {s  —  c). 
2xA 


b  = 


h 


^  =  2x^. 


EXAMPLES. 

1.  A  triangle  has  a  base  of  40  ft.,  and  an  altitude  of 
15  ft. ;  how  many  square  feet  does  it  contain  ? 

2.  Find  the  area  of  a  triangle,  the  length  of  whose  base 
is  25  ft.,  and  the  height  12  ft.  4  in. 

3.  Find  the  number  of  acres  in  a  trianguJar  field  whose 
base  is  20.28  ch.  and  altitude  14.5  ch. 

4.  How  many  acres  are  there  in  a  triangular  lot  whose 
base  is  432  ft.  and  altitude  320  ft.  ? 

5.  Find  the  number  of  hektars  in  a  triangular  field 
whose  base  is  196.8"'  and  altitude  85'". 


MENSURATION. 


303 


6.  Find  the  area  of  a  triangle  whose  sides  are  respec- 
tively 4  ft.,  6  ft.,  and  8  ft. 

7.  Find  the  number  of  hektars  in  a  triangular  field 
whose  sides  are  respectively  62.4",  84.2%  and  106.8™. 

8.  At  60  cents  a  square  yard,  find  the  cost  of  paving  a 
triangular  court  whose  sieves  are  respectively  80  ft.,  75  ft., 
and  60  ft. 

9.  Find  the  altitude  of  a  triangle  whose  area  is  137^ 
sq.  ft.  and  base  20  ft. 

10.  Find  the  altitude  of  a  triangle  whose  area  is  3.25"" 
and  base  502™. 

11.  Find  the  base  of  a  triangle  whose  area  is  20  A.  and 
altitude  80  rd. 

12.  Find  the  base  of  a  triangle  whosp  area  is  12.6*  and 
altitude  30". 

13.  Find  the  area  and  altitude  of  an  equilateral  triangle 
whose  sides  are  each  12  ft.  long. 

14.  Find  the  perpendicular  distances  from  the  vertices 
to  the  opposite  sides  of  a  triangle,  when  the  sides  are  re- 
spectively 12^™  15*=™,  and  20*=". 


Eight  Triangles. 

169.    The  square  of  the  hypotenuse  of  a  right  triangle  is 

equal  to  the  sum  of  the  squares  of 
the  other  two  sides.  This  principle 
is  illustrated  in  the  annexed  dia- 
gram. 

To  find  the  hypotenuse  of  a 
right  triangle  when  the  other  two 
sides  are  given,  extract  the  square 
root  of  the  sum  of  the  squares  of 
the  other  two  sides. 


I    J  ■ 


304  ARITHMETIC. 

To  find  one  side  of  a  right  triangle  when  the  hypotenuse 
and  other  side  are  given,  extract  the  square  root  of  the  differ- 
ence between  the  squares  of  the  hypotenuse  and  the  other  side. 

Note.  The  length  and  breadth  of  a  rectangle  form  two  sides  of  a 
right  triangle  of  which  the  diagonal  of  the  rectangle  is  the  hypotenuse. 

To  find  the  diagonal  of  a  rectangular  parallelopiped,  ex- 
tract the  square  root  of  the  sum  of  the  squares  of  the  three 
dimensions. 

EXAMPLES. 

1.  Find  the  hypotenuse  of  a  right  triangle  whose  base 
is  30  ft.  and  perpendicular  16  ft. 

2.  The  hypotenuse  of  a  right  triangle  is  16:|^  ft.,  and  the 
base  is  15  ft. ;  what  is  the  perpendicular  ? 

3.  The  hypotenuse  of  a  right  triangle  is  3.25",  and  the 
perpendicular  is  3"  ;  find  the  base. 

4.  A  flag-pole  140  ft.  high  casts  a  shadow  105  ft.  in 
length ;  what  is  the  distance  from  the  top  of  the  pole  to 
the  end  of  the  shadow  ? 

5.  What  is  the  length  of  a  ladder  that  will  just  reach 
to  the  top  of  a  house  12"  high,  when  its  foot  is  placed  8.4™ 
from  the  house  ? 

6.  Find  the  height  of  the  eaves  of  a  house  that  can  be 
reached  by  a  ladder  40  ft.  long,  when  the  foot  of  the  ladder 
stands  24  ft.  from  the  house. 

7.  A  pole  was  broken  26  ft.  from  the  bottom,  and  fell 
so  that  the  end  struck  19  ft.  6  in.  from  the  foot ;  find  the 
length  of  the  pole. 

6.  Find  the  width  of  a  street,  from  a  point  in  which  a 
ladder  36  ft.  long  will  reach  a  window  28  ft.  high  on  one 
side^  and  one  25|-  ft.  high  on  the  other, 


MENSURATIOK.  80d 

9.  A  steamer  goes  due  north  at  the  rate  of  15  miles  an 
hour,  and  another  due  west  18  miles  an  hour.  How  far 
apart  will  they  be  in  6  hours  ? 

10.  A  rectangular  field  is  96  rd.  long  and  72  rd.  wide ; 
find  the  length  of  the  diagonal. 

11.  Find  the  longest  straight  line  that  can  be  drawn  on 
a  floor  4.5™  long  and  3.2™  wide. 

12.  The  side  of  a  square  field  is  40  rd. ;  find  the  distance 
between  two  diagonally  opposite  corners. 

13.  The  diagonal  of  a  square  equals  16  ft. ;  find  the 
length  of  a  side. 

14.  What  is  the  length  of  the  diagonal  of  a  room  20  ft. 
long,  16  ft.  wide,  and  12  ft.  high  ? 

15.  Find  the  length  of  the  diagonal  of  a  box  4  ft.  8  in. 
long,  2  ft.  4  in.  wide,  and  7  in.  deep. 

16.  Find  the  diagonal  of  a  cubical  block  whose  edge  is 
3^  inches. 

17.  The  diagonal  of  a  cube  equals  10"=™ ;  find  the  length 
of  an  edge. 

Quadrilaterals. 

170.  A  polygon  of  four  sides  is  called  a  qnadrilateral. 

A  parallelogram  is  a  quadrilateral  whose  opposite  sides 
are  parallel.  A  rectangle  is  a  parallelogram  whose  angles 
are  right  angles ;  a  square  is  a  rectangle  whose  sides  are  all 
equal.  A  rhomboid  is  a  parallelogram  whose  angles  are 
oblique  angles ;  a  rhombus  is  a  rhomboid  whose  sides  are 
all  equal. 

A  trapezoid  is  a  quadrilateral  which  has  two  sides  par- 
allel. 

A  trapezium  is  a  quadrilateral  which  has  no  two  sides 
parallel. 


306 


AKITHMETIC. 


Rkctanglk. 


Square. 


Rhomboid. 


Rhombus. 


TUAPKZDII). 


TRAI'EZIUM. 


The  side  upon  which  a  parallelogram  is  supposed  to  stand 
and  the  opposite  are  called  the  lower  and  upper  bases.  The 
parallel  sides  of  a  trapezoid  are  called  the  bases.  The  per- 
pendicular distance  between  the  bases  of  a  parallelogram  or 
trapezoid  is  the  altitude. 

To  find  the  area  of  any  parallelogram,  multiply  the  base 
by  the  altitude. 

To  find  the  area  of  a  trapezoid,  multiply  half  the  sum  of 
the  parallel  sides  by  the  altitude. 

To  find  the  area  of  a  trapezium,  multiply  the  diagonal  by 
half  the  sum  of  the  perpendiculars  to  it  from  the  vertices  of 
opposite  angles. 

The  area  of  any  polygon  may  be  found  by  dividing  it 
into  triangles  and  obtaining  the  sum  of  their  areas. 


EXAMPLES. 

1.  Find  the  number  of  square  yards  in  a  parallelogram 
whose  base  is  25  ft.  and  altitude  22|-  ft. 

2.  Find  the  number  of  hektars  in  a  parallelogram  whose 
base  is  640^"  and  altitude  180°™. 

3.  Find  the  area  of  a  rhomboid  whose  altitude  is  1,32'" 
and  base  154.4™. 


MENSURATION.  307 

4.  Find  the  area  of  a  rhombus,  of  which  one  of  the 
equal  sides  is  358  ft.,  and  the  perpendicular  distance  be- 
tween it  and  the  opposite  side  is  194  ft. 

5.  The  parallel  sides  of  a  trapezoid  are  30"  and  25.2"*, 
and  the  altitude  is  18.2™ ;  find  the  area. 

6.  Two  sides  of  a  held,  which  are  parallel,  are  respec- 
tively 262  yd.  and  486  yd.,  and  the  perpendicular  distance 
between  them  is  440  yd.     How  many  acres  does  it  contain  ? 

7.  Find  the  area  of  a  trapezium  whose  diagonal  is  21  ft. 
and  the  perpendiculars  to  this  diagonal  are  9  ft.  and  8  ft. 

8.  What  is  the  area  of  a  trapezium,  the  length  of  a 
diagonal  of  which  is  25  ft.,  and  of  the  perpendiculars  from 
the  opposite  vertices  to  the  diagonal  5  ft.  and  17^-  ft.  ? 

Circles. 

171.   For  definitions,  see  §  60. 

The  ratio  of  the  circumference  to  the  diameter  is  the 
same  for  all  circles,  and  it  is  customary  to  represent  this 
ratio  by  the  Greek  letter  tt  (pi). 

The  numerical  value  of  tt  cannot  be  obtained  exactly,  but 
the  value  tt  =  3.1416  is  correct  to  four  decimal  places. 

Note.  ir=3f  is  suflSciently  accurate  for  many  purposes,  and  it  is 
used  to  a  great  extent.    In  this  book  tt  — 3.1416  is  the  value  used. 

To  find  the  circumference  of  a  circle  when  the  diameter 
is  given,  multiply  the  diameter  by  3.1416. 

To  find  the  diameter  of  a  circle  when  the  circumference 
is  given,  divide  the  circumference  by  3.1416. 

To  find  the  area  of  a  circle  when  the  radius  is  given,  mul- 
tiply the  square  of  the  radius  by  3.1416. 

To  find  the  radius  of  a  circle  when  the  area  is  given, 
divide  the  area  by  3.1416,  and  extract  the  square  root  of  the 
quotient. 


308  ARITHMETIC. 

Let  A  represent  the  area  of  a  circle,  C  the  circumference, 
D  the  diameter,  and  B  the  radius  ;  then 


c= 

ttXD 

or 

2X7rXR. 

/)= 

TV 

B  = 

G 

2X7r 

A  = 

■■itXR' 

or 

ix 

7rXi>^, 

Ii  = 

■■^. 

Note.  When  a  circle  is  circumscribed  about  a  square,  the  diagonal 
of  the  square  is  the  diameter  of  the  circle.  When  a  circle  is  inscribed 
in  a  square,  the  diameter  of  the  circle  is  equal  to  a  side  of  the  square. 

EXAMPLES. 

1.  Find  the  circumference  of  a  circle  whose  diameter  is 
22  ft. 

2.  Find  the  circumference  of  a  circle  whose  diameter  is 
SO'". 

3.  Find  the  diameter  of  a  circle  whose  circumference  is 
284*'™. 

4.  Find  the  radius  of  a  circle  whose  circumference  is 
82  ft.  4  in. 

5.  Find  the  diameter  of  a  tree  whose  circumference  is 
25  ft.  10  in. 

6.  Find  the  number  of  acres  in  a  circular  field  whose 
radius  is  32  rd. 

7.  Find  the  number  of  hektars  in  a  circular  field  whose 
radius  is  325™. 

8.  Find  the  area  of  a  circle  whose  diameter  is  2  ft.  7  in. 


MENSURATION.  309 

9.  Find  the  radius  of  a  circle  whose  area  is  163^  sq.  rd. 

10.  Find  the  radius  of  a  circle  whose  area  is  ISGO*** "°. 

11.  Find  the  diameter  of  a  circle  whose  area  is  38  sq.  ft. 

12.  Find  the  circumference  oi^  a  circle  whose  area  is  25'. 

13.  Find  the  number  of  acres  in  a  circular  park  whose 
circumference  is  3^  miles. 

14.  Find  the  diameter  of  a  wheel  which  turns  23  times 
in  going  103.5™. 

15.  How  many  times  will  a  wheel  whose  radius  is  0.762" 
revolve  in  running  1.6043^"  ? 

16.  How  many  turns  per  minute  does  a  pulley  1.3"  in 
diameter  make  when  the  belt  travels  50^'"  per  hour  ? 

17.  A  horse,  tied  to  a  stake,  can  graze  to  the  distance  of 
35  ft.  from  the  stake ;  find  the  number  of  square  yards  of 
surface  on  which  he  can  graze. 

18.  In  a  board  6  ft.  long  and  16  in.  wide  are  two  round 
holes,  one  of  which  is  10  in.  across,  and  the  other  12  in. 
across.     Find  the  area  remaining. 

19.  Find  the  width  of  the  ring  between  two  concentric 
circles  whose  circumferences  are  respectively  225  ft.  and 
300  ft. 

20.  What  is  the  area  of  a  circular  ring  formed  by  two 
concentric  circles  whose  diameters  are  respectively  6  ft. 
4  in.  and  4  ft.  6  in.  ? 

21.  Find  the  area  of  a  circle  inscribed  in  a  square  con- 
taining 225  sq.  ft. 

22.  Find  the  area  of  a  circle  circumscribed  about  a  square 
containing  144*'^". 


310 


ARITHMETIC. 


23.  Find  the  side  of  the  largest  square  that  can  be  laid 
out  in  a  circular  enclosure  whose  diameter  is  10  rd. 

24.  Find  the  side  of  a  square  inscribed  in  a  circle  whose 
areais78.54«'i™ 


Pentagonal  Prism. 


Prisms  and  Cylinders. 

172.  A  prism  is  a  solid  whose  ends,  or  bases,  are  equal 
and  parallel  polygons,  and  whose  sides  are 
parallelograms.  A  prism  is  triangular, 
quadrangular,  pentagonal,  etc.,  according 
as  its  ends  are  triangles,  quadrilaterals, 
pentagons,  etc.  A  right  prism  is  a  prism 
whose  sides  are  perpendicular  to  the  bases. 

A  cylinder  is  a  solid  whose  ends,  or 
bases,  are  circles,  and  whose  lateral  sur- 
face is  a  uniformly  curved  surface.  The 
axis  of  a  cylinder  is  a  straight  line  join- 
ing the  centres  of  the  two  bases.  A  right 
cylinder  is  a  cylinder  whose  axis  is  perpen- 
dicular to  the  bases. 

The  perpendicular  distance  between  the 
bases  of  a  prism  or  cylinder  is  called  the 
altitude. 

To  find  the  lateral  surface  of  a  right 
prism  or  right  cylinder,  multiply  the  perim- 
eter of  a  base  by  the  altitude. 

To  find  the  volume  of  a  prism  or  cylin- 
der, multiply  the  area   of  a  base  by  the 
altitude. 
Let  S  represent  the  lateral  surface  of  a  right  prism  or 
right  cylinder,  V  the  volume  of  any  prism  or  cylinder,  B 
the  area  of  a  base,  P  the  perimeter  of  a  base,  and  H  the 
altitude:  then 


Ctlindkb. 


MENSURATION.  311 

S=FxH. 
V=BxH. 

Let  a  represent  the  radius  of  a  base  of  a  cylinder,  and  D 
the  diameter;  then  the  following  formulae  are  true  for 
cylinders : 

^  =  2X7rX  RXH  OT  ttXDxH. 

V=7r  X  M'  X  H  01  \  X  TT  X  D'  X  H, 

EXAMPLES. 

1.  How  many  square  feet  are  there  in  the  lateral  sur- 
face of  a  right  prism  whose  altitude  is  3  ft.,  and  whose  base 
is  a  regular  hexagon,  each  side  of  which  is  6  in.  long  ? 

2.  The  radius  of  the  base  of  a  cylinder  is  8  in.,  and  the 
altitude  is  2^  ft. ;  how  many  square  feet  are  there  in  the 
lateral  surface  ?  in  the  whole  surface  ? 

3.  The  sides  of  the  base  of  a  triangular  prism  are  re- 
spectively 12, 15,  and  24  feet,  and  the  altitude  is  20  ft. ;  find 

the  cubic  contents. 

4.  Find  the  volume  of  a  prism  whose  base  contains  7^ 
sq.  ft.,  and  the  square  of  whose  height  equals  five  times  the 
number  of  square  feet  in  the  base. 

5.  Find  the  capacity  in  gallons  of  a  cylindrical  cistern, 
measuring  16  ft.  across  and  15  ft.  deep. 

6.  Find  the  number  of  liters  contained  in  a  cup,  measur- 
ing 20"^"^  across  and  31.831*'"  deep. 

7.  Find  the  number  of  cubic  feet  in  a  log  28^  ft.  long 
and  6  ft.  2  in.  round. 

8.  How  many  kiloliters  must  be  drawn  from  a  cylindri- 
cal tank,  the  diameter  of  the  base  being  10™,  in  order  to 
lower  the  surface  7*^  ? 


B12 


ARITHMETIC. 


9.   A  cylindrical  vessel  1"'  high  is  made  of  sheet  iron  2"" 
thick,  and  holds  100\     What  is  its  outer  diameter  ? 

10.  The  diameter  of  a  cylindrical  vessel  filled  with  water 
is  6  in.  An  immersed  stone  displaces  1^  in.  in  depth  of  the 
water.     How  many  cubic  inches  are  there  in  the  stone  ? 


Pyramids  and  Cones. 

173.  A  pyramid  is  a  solid  whose  base  is  a  polygon,  and 
whose  sides  are  triangles  meeting  in  a  common  point,  called 
the  vertex.  A  pyramid  is  triangular,  quad- 
rangular, pentagonal,  etc.,  according  as  its 
base  is  a  triangle,  quadrilateral,  pentagon, 
etc.  A  right  pyramid  is  a  pyramid  whose 
base  is  a  regular  polygon,  and  in  which 
the  perpendicular  from  the  vertex  passes 
through  the  centre  of  the  base. 

A  cone  is  a  solid  whose  base  is  a  circle, 
and  whose  lateral  surface  tapers  uniformly 
to  a  point,  called  the  vertex.  The  axis  of 
a  cone  is  a  straight  line  drawn  from  the 
vertex  to*  the  centre  of  the  base.     A  right 

A  cone  is  a  cone  whose  axis  is  perpendicular 
to  the  base. 
The  altitude  of  a  pyramid  or  cone  is  the 
perpendicular  distance  from  the  vertex  to 
the  base. 
The  slant  height  of  a  right  pyramid  or 
right  cone  is  the  shortest  distance  from  the 
vertex  to  the  perimeter  of  the  base. 
Cone.  A  frustum  of  a  pyramid  or  cone  is  the  part 

of  the  pyramid  or  cone  that  remains  after 
cutting  off  the  upper  part  by  a  plane  parallel  to  the  base. 
The  altitude  of  a  frustum  is  the  perpendicular  distance 


Quadrangular 
Pyramid. 


MENSURATION.  313 

between  the  two  bases,  and  the  slant  height  is  the  shortest 
distance  between  the  perimeters  of  the  bases. 

To  find  the  lateral  surface  of  a  right  pyramid  or  right 
cone,  multiply  the  perimeter  of  the  base  by  one  half  of  the  slant 
height. 

To  find  the  lateral  surface  of  the  frustum  of  a  right  pyra- 
mid or  right  cone,  multiply  one  half  the  sum  of  the  perimeters 
of  the  bases  by  the  slant  height. 

To  find  the  volume  of  a  pyramid  or  cone,  multiply  the  area 
of  the  base  by  one  third  of  the  altitude. 

To  find  the  volume  of  a  frustum  of  a  pyramid  or  cone, 
multiply  the  sum  of  the  areas  of  the  two  bases  and  the  square 
root  of  their  jn'oduct  by  one  third  of  the  altitude. 

Let  S  represent  the  lateral  surface  of  a  right  pyramid  or 
right  cone,  V  the  volume  of  any  pyramid  or  cone,  B  the 
area  of  the  base,  P  the  perimeter  of  the  base,  U  the  alti- 
tude, and  L  the  slant  height ;  then 

S  =  \xPy.L. 
V=^ixBxH. 

Let  E  represent  the  radius  of  the  base  of  a  cone,  and  JJ 
the  diameter;  then  the  following  formulae  are  true  for 
cones : 

S  =  Tr  X  JiX  L  or  ^XttX  D  X  L. 

V=iXTrxR'xH  or  ^\X7rxD'xH. 

The  formulae  for  a  frustum,  representing  the  areas  of  the 
bases  by  B  and  B',  and  the  perimeters  of  the  bases  by  P  and 
P',  are  as  follows  : 

S  =  ix{P+P>)xL. 

V=ix{B-\-B'+  VBxB')  X  H. 

Eepresenting  the  radii  of  the  bases  of  a  frustum  of  a  cone 
by  R  and  R\  and  the^  diameters  by  D  and  D\  the  formulae 
foi  a  frustum  of  a  cone  are 


314  ARITHMETIC. 

S  =  7r  X  {R  -\-  R')  X  L  OT  ^  X  TT  X  (D  -\-  D')  X  L. 
V=  i  X  TT  X  {B'  -h  M"  -\-  E  X  R')  X  H 
0T^X7rX{D'-}-D"4-DxD')xH. 

EXAMPLES. 

1.  How  many  square  feet  are  there  in  the  lateral  sur- 
face of  a  right  pyramid  whose  slant  height  is  6  ft.,  and 
whose  base  is  a  regular  octagon,  each  side  of  which  is  4  ft. 
long? 

2.  The  radius  of  the  base  of  a  right  cone  is  16  in.,  and 
the  slant  height  is  4  ft. ;  how  many  square  feet  are  there  in 
the  lateral  surface  ?  in  the  whole  surface  ? 

3.  The  slant  height  of  a  frustum  of  a  right  pyramid  is 
5™,  and  the  perimeters  of  the  two  bases  are  12™  and  8™  re- 
spectively ;  find  the  lateral  area  of  the  frustum. 

4.  The  slant  height  of  a  frustum  of  a  right  cone  is  8  ft., 
and  the  radii  of  the  bases  are  8  ft.  and  5  ft.  respectively ; 
how  many  square  feet  are  there  in  the  lateral  surface  ?  in 
the  whole  surface  ? 

5.  The  altitude  of  a  pyramid  is  8"",  and  its  base  is  a  rec- 
tangle 3"  by  2" ;  find  the  volume. 

6.  The  altitude  of  a  cone  is  18  ft.,  and  the  radius  of  its 
base  is  6  ft. ;  find  the  volume. 

7.  The  base  of  a  right  triangular  pyramid  is  an  equilat- 
eral triangle,  each  side  of  which  is  6  ft.,  and  the  altitude  is 
9  ft. ;  find  the  cubic  contents. 

8.  Find  the  volume  of  a  pyramid  whose  base  is  5™ 
square,  and  whose  height  equals  the  diagonal  of  the  base. 

9.  Find  the  capacity  in  liters  of  a  pail  25'^'"  deep,  meas- 
uring 28°°  across  the  top  and  18*="  across  the  bottom. 


MENSURATION.  815 


Spheres. 

174.  A  sphere  or  globe  is  a  solid  bounded  by  a  cxirved 
surface,  every  point  of  which  is  equally  distant  from  a 
point  within  called  the  centre.  A 
straiglit  line  passing  through  the 
centre  and  having  its  extremities 
in  the  surface  is  called  a  diameter ; 
a  straight  line  drawn  from  the 
centre  to  the  surface  is  a  radius. 
A  section  of  a  sphere  made  by  a 
plane  passing  through  the  centre 
is  called  a  great  circle,  and  the  Sphere. 

circumference  of  a  sphere  is  the  same  as  a  circumference  of 
a  great  circle. 

To  find  the  surface  of  a  sphere,  multiply  the  circumference 
by  the  diameter. 

To  find  the  volume  of  a  sphere,  multiply  the  surface  by  one 
third  of  the  radius. 

Let  S  represent  the  surface  of  a  sphere,  Fthe  volume,  R 
the  radius,  and  D  the  diameter  ;  then 

>S'  =  4X7rXi?^  or  ^XT)*. 
F=  f  X  TT  X  72"  or  ^  X  TT  X  D*. 

EXAMPLES. 

1.  Find  the  number  of  square  feet  in  the  surface  of  a 
sphere  whose  radius  is  8  ft. 

2.  Find  the  number  of  cubic  feet  in  the  volume  of  a 
sphere  whose  radius  is  6  ft. 

3.  How  many  cubic  centimeters  are  there  in  a  cannon 
ball  whose  diameter  is  IS'^"  ? 


31^  ARITHMETIC. 

4.  How  many  square  inches  of  leather  will  cover  a  ball 
8  in.  in  circumference  ? 

5.  A  ball  contains  2144.6656  cu.  in. ;  what  is  the  diam- 
eter? 

6.  The  earth's  surface  contains  about  509294630'"^'"; 
find  the  radius. 

7.  A  hemispherical  vessel  measures  2^  ft.  across  the 
top ;  how  many  gallons  does  it  hold  ? 

8.  A  spherical  shell  of  copper  has  an  outer  radius  of  2™ 
and  is  5^™  thick.  What  is  the  weight  of  this  shell  in  kilo- 
grams when  it  is  filled  with  mercury,  the  specific  gravity 
of  the  copper  being  8.8  and  mercury  13.6  ? 

Similar  Surfaces  and  Solids. 

175.  Surfaces  or  solids  which  have  the  same  form  are 
said  to  be  similar. 

Like  dimensions  of  similar  surfaces  or  similar  solids  are 
jwoportional. 

The  areas  of  similar  surfaces  are  to  each  other  as  the  squares 
of  their  corresponding  dimensions. 

The  volumes  of  similar  solids  are  to  each  other  as  the  cubes 
of  their  corresponding  dimensions. 

I.  A  triangle  whose  base  is  12  ft.  has  an  area  of  54  sq. 
ft. ;  find  the  base  of  a  similar  triangle  whose  area  is  96 
sq.  ft. 

54  :  96  :  :  12^  .-  a^.  Let  x  represent  the  base  of  the  second 

\Q      A        ^  triangle.     Since  the  areas  are  to  each  other 

S^XZ^XX^     OKA         ^^  ^'"^  squares  of  their  corresponding  dinun- 

~  "m  ~  sions,  54  :  96  :  :  12^  :  x^.      From   this  propor- 

?  tion  we  find  that  a-2  =  256.     Hence  the  base 

^  of  tlie  triangle  is  the  square  root  of  256,  or 

Ans.  16  ft.  16  ft. 


MENSURATION.  317 

II.   A  cylinder  which  is  9  ft.  high  contains  504  cu.  ft. ; 
find  the  volume  of  a  similar  cylinder  6  ft.  high. 

Let  X  represent  the  volume 


of  the  second  cylinder.  Since 
the  volumes  are  to  each  other 
as  the  cubes  of   their  corre- 


504  :  « : :  9^ :  6^ 
66 

T6{8      2      2      2 
^     K2<iXi>if-448_i4Qip„  ft      spo^'ling  'Ji'nensions, 

2     2     Q 

1*     y     ^  From  this  proportion  we  find 

that  a:=149|. 

EXAMPLES. 

1.  The  bases  of  two  similar  triangles  are  6  ft.  and  8  ft. 
respectively,  and  the  altitude  of  the  former  is  9  ft. ;  find 
the  altitude  of  the  latter. 

2.  The  hypotenuse  of  a  right  triangle  is  26";  find  the 
hypotenuse  of  a  similar  triangle  which  contains  twice  the 
area. 

3.  The  area  of  a  trapezoid  is  108  sq.  ft.,  and  its  altitude 
is  6  ft. ;  find  the  altitude  of  a  similar  trapezoid  whose  area 
is  192  sq.  ft. 

4.  How  many  circles,  each  4  in.  in  diameter,  will  equal 
in  area  a  circle  whose  diameter  is  2  ft.  ? 

5.  Two  farms  of  exactly  similar  form  contain  respec- 
tively 16  and  25  acres.  One  side  of  the  former  is  60  rd.  in 
length ;  find  the  corresponding  side  of  the  latter. 

6.  If  a  cistern  can  be  filled  in  30  min.  by  a  pipe  1  in.  in 
diameter,  in  what  time  can  it  be  filled  by  a  pipe  3  in.  in 
diameter  ? 

7.  If  a  pyramid  6  ft.  high  contains  45  cu.  ft.,  what  is 
the  height  of  a  similar  pyramid  that  contains  100  cu.  ft.  ? 

8.  How  many  spheres,  each  6  in.  in  diameter,  will  equal 
in  volume  a  sphere  whose  diameter  is  2  ft.  ? 


318  ARITHMETIC. 

9.  If  a  man  digs  a  small  square  cellar,  measuring  6  ft, 
each  way,  in  one  day,  how  long  would  it  take  him  to  dig  a 
similar  one  measuring  10  ft.  each  way  ? 

10.  If  a  stack  of  hay  5  ft.  high  weighs  100  lb.,  find  the 
weight  of  a  similar  stack  24  ft.  high. 

11.  If  a  rope  1  in  in  diameter  weighs  2^  lb.,  what  is  the 
diameter  of  a  rope  of  the  same  length  which  weighs  50  lb.  ? 

12.  How  far  from  the  base  must  a  cone  whose  altitude  is 
8  ft.  be  cut  off  so  that  the  ;Prv*-;t'\m  .shaP  bp  equivalent  to 
one  half  of  the  cone  ? 


MISCELLANEOUS  EXAMPLES.  319 


MISCELLANEOUS   EXAMPLES. 

1.  Divide   3380321   by  MDCCXCIX,  and  express  the 
quotient  by  the  lioman  system  of  notation. 

2.  Find,  by  casting  out  the  nines,  whether  the  following 
is  correct :  349761  x  28637  =  10015819397. 

3.  Multiply  4.32  by  0.00012. 

4.  Divide  0.002268  by  10.8. 

5.  Divide  the  product  of  12,  20,  and  30  by  the  product 
of  15,  24,  and  18,  by  cancellation. 

6.  Find  the  factors  and  the  greatest  common  divisor  of 
1498,  1582,  and  2331. 

7.  Arrange  in  order  of  magnitude  J^,  |^,  and  -J-J. 

8.  Eeduce  ^^,  -^j   and  \^  to  their  least  common 
denominator. 

9.  Divide  |  of  47  by  ^  of  51. 

10.  Find  the  value  ofJ-J  +  4f+29+3^;  reduce  the 
result  to  its  lowest  terms,  and  also  to  a  decimal  form. 

11.  At  f  1.75  a  rod,  what  will  it  cost  to  fence  a  piece  of 
ground  63.5  rd.  long  and  27.75  rd.  wide  ? 

12.  From  a  piece  of  cloth  containing  84f  yd.  there  were 
sold  4f  yd.,  26 J  yd.,  and  \  of  7J  yd. ;  how  much  remains  ? 

13.  Name  all  the  prime  numbers  in  the  series  of  numbers 
between  1  and  30  inclusive ;  resolve  all  the  composite  num- 
bers into  their  prime  factors ;  and  name  all  the  perfect 
S(juares,  cubes^  and  other  powers  in  the  same  series. 


320  ARITHMETIC. 

14.  How  mucli  will  be  paid  for  3760  lb.  of  coal  at  f  15 
a  ton  ? 

15.  Pind  the  product  of  157.757  and  15.3254  to  two  places 
of  decimals. 

16.  Divide  1728  by  0.00144,  and  multiply  the  result  by 
0.000012. 

17.  Divide  $125  among  4  boys  and  3  girls,  and  give  each 
boy  "I  as  much  as  each  girl. 

18.  Bought  360  gallons  of  wine  at  $2.60  a  gallon ;  paid 
for  carriage  |17.20,  and  for  duties  $86.50.  If  ^  of  it  be 
lost  by  leakage,  at  what  price  must  the  remainder  be  sold 
to  gain  $50  on  the  whole  transaction  ? 

19.  Find  the  product  of  three,  three  hundredths,  thirty- 
three  thousandths,  three  thousand  millionths,  and  two 
twenty-fifths. 

20.  Divide  ten  thousand  six  hundred  twenty-five  bill- 
ionths  by  seventeen  thousandths,  and  extract  the  square 
root  of  the  quotient. 

21.  Find  the  value  of  the  following  fraction  to   three 
1.0045  X  0.0875 


decimal  places 


0.0016 


22.  Find  the  sum  of  five,  five  tenths,  thirty-seven  thou- 
sandths, one  thousand  millionths,  XIX,  MDCCCLXXXI, 
and  O.iS. 

1.28 

23.  Reduce ^1 j-  3  to  its  simplest  decimal  form. 

24.  Reduce  3.36  inches  to  a  decimal  fraction  of  a  rod. 

25.  Reduce  a  pressure  of  22.5  lb.  Avoirdupois  per  square 
foot  to  ounces  per  square  inch. 


MISCELLANEOUS    EXAMPLES.  321 

26.  If  either  5  oxen  or  7  horses  will  eat  up  the  grass  of 
a  field  in  87  days,  in  what  time  will  2  oxen  and  3  horses  eat 
up  the  same  *? 

27.  How  many  liters  of  water  may  be  contained  in  a 
reservoir  10*"  long,  6™  wide,  and  4™  high  ?  What  will  be 
the  weight  in  kilograms  ? 

28.  A  bin  is  2^""  high,  and  contains  IG^^'.  The  base  of 
the  bin  is  a  square ;  how  many  centimeters  are  there  in  one 
of  its  sides  ? 

29.  If  4  men  can  mow  15  acres  in  5  days  of  14  hours, 
in  how  many  days  of  13  hours  can  7  men  mow  19J  acres  ? 

30.  I  buy  300  bu.  of  grain  consisting  of  wheat,  rye,  and 
oats,  in  the  proportion  of  3,  4,  and  5.  How  many  bushels 
of  each  do  I  buy  ? 

31.  The  sum  of  two  numbers  is  100,  and  i  of  one  of  them 
is  f  of  the  other ;  find  the  numbers. 

32.  How  much  water  must  be  mixed  with  31  gal.  of 
another  liquid  which  cost  $45.25,  that  the  mixture  may  be 
sold  at  $1.25  per  gallon,  and  25^0  be  gained? 

33.  What  is  the  interest  on  $647.65  for  2  yr.  5  mo.  10  da. 
at  5%  ? 

34.  What  sum  will  produce  $12.50  interest  in  20  days 

at  4%  ? 

35.  Find  the  selling  price  of  goods  by  which  there  is  a 
loss  of  2%  and  an  actual  loss  of  $54.50. 

36.  If  12  barrels  of  corn  will  pay  for  10  cords  of  wood, 
and  48  cords  of  wood  will  pay  for  8  tons  of  hay,  and  5  tons 
of  hay  will  pay  for  16  kegs  of  nails,  how  many  barrels  of 
corn  will  pay  for  12  kegs  of  nails  ? 

37.  Find  the  reciprocal  of  155  carried  to  five  decimal 
places. 


322  ARITHMETIC. 

38.  Convert  into  a  decimal  i-±^  x  0.00025. 

■  0.075 

39.  What  are  the  prime  factors  of  1716?  How  many 
integral  divisors  has  this  number,  and  what  are  they  ? 
What  is  the  smallest  integer  by  which  this  number  can  be 
multiplied,  so  that  the  product  shall  be  a  square  ? 

40.  Reduce  to  its  simplest  form  the  expression  -  of  — ^ 

41.  Reduce  5f  and  10^  to  the  decimal  form,  and  divide 
the  first  by  the  second. 

42.  Find  the  greatest  common  divisor  of  26^,  28|^,  and 
291 

43.  Find  the  least  common  multiple  for  the  numbers  J, 
2.1,  5.25,  and  f. 

44.  Change  0.013  to  an  equivalent  fraction  whose  denom- 
inator is  135. 

45.  -^  of  f  of  f  ft.  equals  what  decimal  of  a  rod  ? 

46.  A  block  of  stone  (sp.  gr.  2.5)  is  1™  long,  S**'"  wide, 
and  45'^™  thick.     How  many  kilograms  does  it  weigh  ? 

47.  rind  the  number  of  liters  in  a  vat  2™  by  75'='"  by  50'=". 
Also  find  the  weight  in  kilograms  of  the  sulphuric  acid 
(sp.  gr.  1.84)  required  to  fill  it. 

48.  If  a  meter  is  39.37  inches,  how  many  feet  are  there 
in  a  dekameter  ?  How  many  square  centimeters  in  a  square 
kilometer  ? 

49.  If  4  masons  build  27  yd.  of  wall  in  5  days  working 
9  hr.  a  day,  in  how  many  days  will  32  masons  build  81  yd. 
of  a  similar  wall  if  they  work  10  hr.  a  day  ? 


MISCELLANEOUS   EXAMPLES.  323 

50.  Separate  772|  into  three  numbers,  which  shall  be  in 
the  same  proportion  as  2|,  -^f  and  y^^. 

51.  Compute  the  square  of  the  sum  of  the  cubes  of  the 
first  twelve  prime  numbers,  and  check  all  the  work  by  cast- 
ing out  the  nines. 

52.  Multiply  34.056  by  0.065043,  obtaining  the  product 
to  four  decimal  places. 

53.  Divide  0.0144  by  4800;  multiply  the  quotient  by 
6.004,  and  extract  the  square  root  of  the  product. 

54.  Simplify  ^  of  -^i^ ,  and  divide  the  result  by  0.0018. 

li  X  3^ 

55.  Find  the  least  common  multiple  of  76,  105,  150,  and 
175. 

56.  Reduce  to  a  common  denominator  and  add  f  X  J  X  |, 
A,  f  ,  and  -^. 

57.  If  f  of  a  cord  of  wood  cost  $3.33J,  what  would  }  of 
a  cord  cost  ? 

58.  75  miles  equals  how  many  kilometers  ?  75  pounds 
Avoirdupois  equals  how  many  kilograms  ?  75  quarts  equals 
how  many  liters  ? 

59.  A  man's  height  is  174'=™.  What  is  his  height  in  feet 
and  inches  ? 

60.  Find  the  value  of  17'  of  sulphuric  acid  (sp.  gr.  1.84) 
at  5  cents  a  kilogram. 

61.  By  selling  a  horse  for  $64.75,  I  lost  7^%;  what  per 
cent  would  I  have  gained  by  selling  him  for  $73.50  ? 

62.  Sold  steel  at  $25.44  a  ton  with  a  profit  of  6%  and  a 
total  profit  of  $103.32.     What  quantity  was  sold  ? 

63.  If  I  buy  stocks,  par  value  $187.50,  at  15%  below  par 
and  sell  them  at  19^%  above  par,  what  is  the  gain  per  cent 
on  my  investment  ? 


324  ARITHMETIC. 

64.  If  I  buy  coal  at  $4.12  per  ton  on  six  months'  credit, 
for  what  must  I  sell  it  immediately  to  gain  10%  ? 

65.  Find  the  amount  of  f  342.42  from  Feb.  5th,  1879  to 
Mar.  15th,  1881,  with  interest  at  7%,  and  reduce  it  to  pounds 
sterling. 

66.  Find  the  interest,  discount,  and  bank  discount  on 
$25  for  60  days  at  7%. 

67.  Find  the  interest,  discount,  and  bank  discount  on 

$17.50,  due  in  30  days,  at  4|-%. 

68.  A  merchant  bought  flour  for  $1000  cash  and  sold  the 
same  immediately  for  $1200  on  6  mo.  credit,  for  which  he 
received  a  note.  If  he  should  get  the  note  discounted  at  a 
bank  at  5%,  what  would  be  the  gain  on  the  flour  ? 

69.  A  and  B  can  do  a  piece  of  work  in  4  hours,  A  and  C 
in  3f  hours,  and  B  and  C  in  5|  hours.  In  what  time  can  A 
do  it  alone  ? 

70.  Divide  52  into  such  parts  that  \  of  one  part  shall 
equal  |  of  the  other. 

71.  A  cubical  cistern  holds  1331^^  of  water ;  what  is  the 
length  of  an  inner  edge  ? 

72.  Arrange  in  order  of  magnitude  |-|,  4|,  and  0.89. 

73.  Show  that  the  square  root  of  0.3  lies  between  |^ 
and  -fj. 

74.  I  have  a  rectangular  lot  of  land,  64  rd.  long  and  36 
rd.  wide,  and  a  square  lot  of  the  same  area ;  how  many  more 
feet  of  fencing  will  be  needed  for  the  former  lot  than  for 
the  latter  ? 

75.  If  144  pounds  Avoirdupois  be  equivalent  to  175 
pounds  Troy,  what  is  the  ratio  of  the  pennyweight  Troy  to 
the  dram  Avoirdupois  ? 


MISCELLANEOtTS  EXAMPLES.  325 

76.  Reduce  to  equivalent  fractions  having  a  common 
denominator  |  of  |,  2f,  5|,  and  |  of  ^  of  3J. 

77.  The  number  209.069673692836  is  composed  of  three 
factors,  of  which  two  are  20083.6  and  0.260075  j  find  the 
third  factor. 

78.  Simplify  H  ><  3|  of  2^, 

l  +  H-iJ 

79.  Simplify  (3.2  +  0.004  -  t.lll)x  0.25, 

^    -^  (4 -^  0.2) -17.907 

80.  Simplify 1__. 

2+     ^^ 


4  +  ? 
6 


81.  Write  in  Arabic  numerals  the  value  of  the  expression 
[MDCCCLXXXIII  ^  16.6]  x  [(2.5  - 1.25)  --  0.03]. 

82.  Reduce  |^  to  its  lowest  terms  ;  reduce  the  result  to 
a  decimal,  and  extract  the  square  root  to  three  figures. 

83.  What  is  the  length  in  meters  and  decimeters  of  a  side 
of  a  square  which  contains  0.1335*? 

84.  Find  the  length  in  dekameters  of  the  side  of  a  square, 
the  area  of  which  equals  the  area  of  a  rectangle  which  is 

]^Km  gin  loj^g  and  4i|Hm  ^^^Jg^ 

85.  A  square  field  contains  1016064  sq.  ft.  What  is  the 
length  of  a  side  expressed  in  meters  ? 

86.  Find  the  side  to  millimeters  of  a  cubical  box  that 
contains  1^™. 

87.  The  volume  of  a  sphere  is  0.056  cu.  yd.  What  is  the 
length  in  inches  of  the  side  of  a  cube  containing  the  same 
volume  ? 

88.  Find  the  edge  of  a  cubical  can  which  will  hold  27.57^» 
of  sulphuric  acid,  whose  specific  gravity  is  1.8. 


826  ARITHMETIC. 

89.   Find  a  fourth  proportional  to  0.37,  8.9,  and  4.3,  and 
extract  the  cube  root  of  it  to  two  decimal  places. 


90.  Find  the  fourth  term  in  V4.913  :  0.0016  : :  48000  :  . 

91.  If  60  cannon  firing  5  rounds  in  8  min.  kill  350  men 
in  75  min.,  how  many  cannon  firing  7  rounds  in  9  min.  will 
kill  980  men  in  25  min.  ? 

92.  A  man  travelled  2  days  at  the  rate  of  15  miles  per 
day,  4  days  at  the  rate  of  20  miles  per  day,  and  5  days  at 
the  rate  of  30  miles  per  day ;  what  was  his  average  rate  of 
travel  per  day  ? 

93.  A  farmer  divides  among  his  3  sons  246  A.  1  E.  32  P., 
sharing  it  among  them  as  the  numbers  3,  4,  and  5 ;  what 
were  the  shares  ? 

94.  How  many  kilograms  are  there  in  a  cubic  foot  of 
water  ? 

95.  A  rectangular  box  is  4™  long,  30^""  wide,  and  20*"  deep. 

(i)   Find  its  capacity  in  liters, 
(ii)   What  weight  of  water  will  it  contain  "? 
(iii)    What  weight  of  mercury  (sp.  gr.  13.6)  will  it  contain  ? 

96.  An  empty  bottle  weighs  380^;  when  filled  with  water 
it  weighs  0.985^^.     How  many  liters  does  the  bottle  hold  ? 

97.  Find  the  present  worth  of  a  note  for  $1320,  due  in 
Syr.  4  mo.,  money  being  worth  6%.  Find  also  what  could 
be  obtained  for  the  same  note  at  bank  discount. 

98.  Find  the  interest,  discount,  and  bank  discount  on 
165.33  for  90  days  at  41%. 

99.  What  is  the  difference  between  the  true  and  bank 
discount  of  $250,  due  10  mo.  hence,  at  7%  ? 

100.  How  long  must  a  note  of  $243  at  3|%  run  that  its 
interest  may  equal  the  interest  on  a  note  of  $125  for  7  mo. 


MISCELLANEOUS  EXAMPLES.  327 

101.  How  many  times  does  the  least  common  multiple  of 
6,  25,  40,  and  75  contain  the  square  of  their  greatest  common 
divisor  ? 

102.  Reduce  -^  and  ^^^  to  their  least  common  denom- 
inator ;  add  the  results,  and  express  the  sum  decimally  to 
four  places. 

103.  Divide  (^-jIt)  ^^7  (A  of  0.00616),  carrying  the 
quotient  to  five  places  of  decimals. 

104.  Simplify  ^+^^  +  (|^'. 

is)  ~1 

105.  4ill^+(3-2J)-(4-3i)   equals   what?      Ex- 

?  "^   T 
tract  the  square  root  of  the  result  to  two  decimal  places. 

106.  Find  the  value  of  (^^  of  —^  divided  by  ^,  and 
extract  the  square  root  of  the  quotient  to  two  decimal  places. 

107.  Simplify  the  expression  J^^^x^- 

108.  How  many  rods  of  fence  will  it  take  to  enclose  a 
20  acre  lot  in  the  form  of  a  square  ? 

109.  If  a  man  can  walk  16  rods  in  |  of  a  minute,  in  what 
time  can  he  walk  0.00164  of  a  mile  ? 

110.  The  length  of  a  rectangular  field  containing  30  acres 
is  3  times  its  width.     Find  the  length  of  the  field  in  feet. 

111.  A  cubic  inch  of  gold  is  hammered  out  until  it  covers 
6  acres ;  how  many  leaves  of  gold  of  this  thickness  would  it 
take  to  make  one  inch  ? 

112.  A  cube  contains  79507  cu.  in.     How  many  square 

inches  does  its  surface  contain? 

113.  How  many  rods  of  fence  will  be  required  to  enclose 
640  acres  of  land  in  a  square  form  ? 


828  ARITHMETIC. 

114.  A  man  lost  ^,  ^,  and  f  of  his  money,  and  then  had 
^2600  left ;  what  sum  had  he  originally,  and  how  much  per 
cent  had  he  lost  ? 

115.  Eeduce  J  of  6%  of  1.05 -j-^\  of  -f^  to  the  simplest 
form. 

116.  I  earned  $10  by  collecting  bills  on  which  a  discount 
of  10%  was  allowed  for  cash.  My  commission  was  5%  ;  how 
much  did  I  collect  ? 

117.  If  4:^%  Government  bonds  sell  at  116,  what  sum  of 
money  invested  in  them  will  yield  an  interest  of  f  1  per 
day? 

118.  Bought  a  bill  of  goods  on  6  months'  credit  for  $500. 
What  would  be  the  gain  per  cent  on  my  bargain  if  I  sold 
the  same  at  once  for  $525  cash,  interest  being  reckoned  at 
6%  per  annum  ? 

119.  Find  the  difference  between  the  true  and  bank  dis- 
counts on  a  note  for  $1000,  due  3  mo.  hence,  money  being 
worth  6%. 

120.  Find  the  interest,  discount,  and  bank  discount  on 
$327.19  for  90  days  at  71%. 

121.  Bought  $1500  worth  of  goods,  half  on  6  months'  and 
half  on  9  months'  credit.  What  sum  at  7%  interest,  paid 
down,  would  discharge  the  bill  ? 

122.  Find  the  principal  that  will  amount  to  $962  in  4  yr. 
6  mo.  at  41%. 

123.  Find  the  compound  interest  on  $300  for  2  yr.  at  4%, 
interest  being  compounded  semi-annually. 

124.  Find  the  annual  interest  of  $200  for  3  yr.  1  mo.  at 
6%. 

125.  Find  the  greatest  common  divisor  of  113.355  and 
3.141592. 


MISCELLANEOUS   EXAMPLES.  329 

126.  Divide  2|  +  |  by  |1. 

127.  Simplify  5:52^  +  ^^,   expressing    the   result    in 
decimal  form. 

128.  [5|  X  tt  X  f +1.0176]  -5-  [3f  +  300.003]  equals  what  ? 

129.  Keduce  to  a  decimal  form    ^  ,  and  from  it  subtract 
0.01  of  |.  ^'^^^ 

130.  Eeduce  to  a  vulgar  fraction  the  decimal  0.0001234. 
Test  your  answer  by  reversing  the  process. 

131.  Find  the  square  root,  to  three  places  of  decimals,  of 
15.75 


10h-{- 


1.5 


4  of  2i 

132.  Find  the  sum  of  3^,  6f  8^2^,  and  65|,  reduce  the 
fractional  part  to  a  decimal,  and  extract  the  cube  root  of 
the  result. 

133.  A  and  B,  44  miles  apart,  travel  towards  each  other. 
A  travels  -^j  of  the  whole  distance,  while  B  travels  ^  of  the 
remainder.     How  far  are  they  then  apart  ? 

134.  Two  engines,  40  miles  apart,  approach  each  other  at 
the  rate  of  25  and  35  miles  an  hour.  Find  the  time  and 
place  of  their  meeting. 

135.  A  river  10™  deep  and  |^™  wide  flows  2^*"  an  hour ; 
find  the  number  of  kiloliters  of  water  that  falls  into  the  sea 
in  a  minute ;  also  its  weight  in  kilograms. 

136.  If  a  sheet  of  paper  weighs  8^^  per  square  meter,  find 
the  weight  in  grams  of  a  piece  IJ"*  long  and  25'^^  wide. 

137.  What  is  the  difference  in  volume  between  two  blocks 
of  granite,  one  1™  long,  6*^"  wide,  and  0.5"  thick,  the  other 
300°"  long,  40«°  wide,  and  0.2"  thick  ? 


330  ARITHMETIC. 

138.  Find  tlie  weight  in  kilograms,  and  in  pounds,  of  a 
rectangular  block  of  marble  (sp.  gr.  2.83)  3.7"'  long,  7*^"' wide, 
and  30'""  thick. 

139.  If  I  buy  macaroni  at  30  cents  a  kilo,  pay  f  12  a 
metric  ton  for  transportation,  and  sell  at  14  cents  a  pound, 
what  per  cent  do  I  gain  or  lose  ? 

140.  A  square  field  contains  0.08346  of  an  acre.  Find  the 
length  of  one  side  of  the  field  in  meters,  the  hektar  being 
equal  to  2.4711  acres. 

141.  The  stere  contains  1.308  cu.  yd.  How  many  meters 
are  there  in  the  side  of  a  cube  containing  0.056  cu.  yd.  ? 

142.  If  35  men  can  build  a  wall  50  ft.  long,  2  ft.  thick, 
and  10  ft.  high  in  8  days,  how  long  will  it  take  50  men  to 
build  a  wall  250  ft.  long,  3  ft.  thick,  and  7  ft.  high  ?  If  the 
first  wall  costs  $910,  what  will  the  second  one  cost  ? 

143.  How  many  men  would  be  required  to  cultivate  a 
field  of  2|-  acres  in  5^  days  of  10  hours  each,  if  each  man 
completed  77  square  yards  in  9  hours  ? 

144.  An  estate  is  divided  among  three  persons.  A,  B,  and 
C,  so  that  A  has  f  of  the  whole,  and  B  has  twice  as  much  as 
C.  It  is  found  that  B  has  27  acres  more  than  C.  How 
large  is  the  estate  ? 

145.  Copper  weighs  550  pounds,  and  tin  462  pounds  to 
the  cubic  foot.  What  will  be  the  weight  of  a  cubic  foot  of 
a  mixture  6  parts  copper  to  5  parts  tin  ? 

146.  A  man  bought  16  horses  and  19  cows  for  f  1865. 
He  paid  upon  the  average  yV  as  much  for  a  cow  as  he  did 
for  a  horse.  What  was  the  average  price  per  head  he  paid 
for  the  horses  ? 

147.  By  selling  a  lot  of  land  for  $783  Host  13%.  What 
would  it  have  brought  if  I  had  sold  it  at  a  loss  of  8^%  ? 


MISCELLANEOUS   EXAMPLES.  331 

148.  At  what  rate  per  cent  is  the  deduction  made  when 
19  s.  10^  d.  is  taken  from  an  account  of  £39  15  s.  in  con- 
sideration of  immediate  payment  ? 

149.  At  what  per  cent  premium  must  a  4%  perpetual 
bond  be  bought  in  order  that  it  may  pay  only  3^%  on  the 
investment  ? 

150.  Find  the  bank  discount  of  a  note  for  $25000  for 
2yr.  6  mo.  at  3^%. 

151.  Find  the  principal  that  will  amount  to  $724.92  in 
2yr.  3  mo.  at  3J%. 

152.  Find' the  simple  interest,  the  annual  interest,  and 
the  compound  interest  of  $1200  for  2  yr.  6  mo.  18  da.  at  4%. 

153.  What  is  the  interest,  discount,  and  bank  discount  on 
$127.42  for  65  days  at  5%  ? 

154.  Find  the  difference  between  the  amount  of  $1000 
for  3yr.  at  6%  compounded  yearly,  and  at  3%  compounded 
half  yearly. 

155.  Find  the  square  root  of  five  million  five  thousand 
and  five  tenths  to  two  decimal  places. 

156.  Compute  the  value  of  3  +  V3  +  ■v^29  to  three  places 
of  decimals. 

157.  Multiply  the  square  root  of  0.173056  by  the  cube 
root  of  iff  If. 

158.  What  is  the  difference  between  the  square  root  and 
the  cube  root  of  1771561  ? 

159.  On  a  map  whose  scale  is  -^  of  an  inch  to  a  mile, 
what  would  be  the  area  covered  by  a  tract  of  land  contain- 
ing  720  square  miles  ? 

160.  A  rectangular  tank,  with  a  square  base,  3  ft.  deep, 
contains  675  cu.  ft.     Find  the  length  of  a  side  of  the  base. 


332  AETTHMETIC. 

161.  A  man  paints  two  sides  of  a  wall  7  ft.  high  in  31  hr. 
6  min.  40  sec.  If  he  can  paint  4  sq.  yd.  in  an  hour,  how 
long  is  the  wall  ? 

162.  A  certain  square  field  contains  38.75  acres.  Com- 
pute the  length  of  one  side  of  the  field  in  meters.  (Given 
1*1'"  =  1550  sq.  in.) 

163.  The  specific  gravity  of  iron  is  7.2;  find  the  volume 
in  cubic  decimeters,  and  the  weight  in  kilograms,  of  a  block 
of  iron  whose  dimensions  are  5,  8,  and  11  inches. 

164.  A  railroad  train  makes  a  mile  in  57  seconds.  What 
is  its  rate  per  hour,  and  what  per  cent  of  the  hour  is  occu- 
pied in  its  making  a  single  mile  ? 

165.  If  5  horses  eat  as  much  as  6  oxen,  and  12  oxen  eat 
12  tons  of  hay  in  40  days,  how  much  hay  will  7  horses  and 
15  oxen  eat  in  65  days  ? 

2      \2^      7-1-      6/ 

166.  Express  the  value  of -^-^ — :— i  exactly  as 

a  decimal. 

167.  Which  is  the  larger,  |^  or  VJ  ? 

5j  +  |i- 0.725 

168.  Eeduce  .   ,  ^  .^ to  an  equivalent  decimal. 

4  +  3.45  ^ 

169.  Simplify  14  of  24  +  6J  -5-  2}  +  (^54  +  0-24 +  0.53^ 

V  2.2-0.64; 

170.  From  the  sum  of  3|  and  4f  subtract  6f-,  multiply 
the  difference  by  |  of  fj  of  88,  and  find  what  fraction  the 
product  is  of  999. 

171.  A  cistern  6.84"'  long  and  2.36™  wide  contains  34^^  of 
wine.     What  is  the  depth  of  the  liquid  ? 

172.  How  many  cubic  meters  are  there  in  a  cord  ? 


MISCELLANEOUS  EXAMPLES.  333 

173.  ixovv  many  liters  of  water  are  there  in  a  full  rectan- 
gular tank  12  ft.  long,  6  ft.  wide,  and  4  ft.  deep  ? 

174.  A  tank  which  holds  100  gal.  can  be  filled  by  one 
pipe  in  25  min.,  and  emptied  by  another  pipe  in  45  min. ;  if 
both  are  opened  together,  how  long  will  it  take  to  fill,  and 
how  much  water  will  have  been  lost  ? 

175.  If  8  men  can  build  a  brick  wall  125  ft.  long,  2  ft. 
wide,  and  4  ft.  high,  in  4  days,  working  10  hr.  each  day, 
how  many  days  will  it  take  12  men  to  build  a  wall  465  ft. 
long,  3  ft.  wide,  and  6  ft.  high,  working  8  hr.  each  day  ? 

176.  Two  men  undertake  to  do  a  piece  of  work  for  $6. 
One  could  do  it  alone  in  5  days,  and  the  other  in  8  days. 
With  the  assistance  of  a  boy,  they  finish  it  in  2J  days.  How 
should  the  money  be  divided  ? 

177.  A  gallon  contains  231  cu.  in.,  and  a  bushel  2150.4 
cu.  in. ;  how  will  a  liquid  quart  compare  with  a  dry  quart  ? 

178.  What  is  the  present  worth  of  $1000,  due  6  mo.  hence, 
money  being  worth  6%  ? 

179.  Find  the  interest,  discount,  and  bank  discount  on 
$416.03  for  60  days  at  7^%. 

180.  What  is  the  difference  between  the  true  discount 
and  that  taken  by  banks  on  $1500,  due  one  year  hence  with- 
out grace  ?     The  rate  of  discount  in  both  cases  is  5%. 

181.  What  is  the  difference  between  the  simple  and  com- 
pound interest  on  $700  for  2  yr.  6  mo.  at  7%,  interest  com- 
pounded annually  ? 

182.  A  note  for  $500  at  60  days  without  interest  is 
bought  for  $450.  What  is  the  profit  if  money  is  worth  1% 
a  month  ? 

183.  How  many  rods  of  fence  will  it  take  to  enclose  a 
square  field  containing  exactly  one  acre  ? 


334  AEITHMETIC. 

184.  The  area  of  a  circle  is  5  sq.  rd.  What  is  the  length 
in  inches  of  one  side  of  a  square  which  contains  the  same 
area? 

185.  Find  the  depth  in  meters  of  a  cubical  cistern  which 
has  a  capacity  of  300001  Give  the  result  to  three  decimal 
places. 

186.  A  is  156  miles  ahead  of  B.  A  travels  30  and  B  42 
miles  a  day.     In  how  many  days  will  B  overtake  A  ? 

187.  If  4  men  or  6  boys  can  do  a  piece  of  work  in  27^ 
days,  in  how  many  days  will  5  men  and  9  boys  do  it  ? 

188.  If  8  horses  consume  3|-  tons  of  hay  in  30  days,  how 
long  will  4^  tons  last  10  horses  and  15  cows,  each  cow  con- 
suming I  as  much  as  a  horse  ? 

189.  A  carriage,  at  the  rate  of  8^  miles  an  hour,  coni 
pletes  "I  of  a  certain  distance  in  3^  days ;  in  how  many  days 
will  it  complete  ^  of  the  same  distance,  going  at  the  rate  of 
10  miles  an  hour  ? 

190.  There  are  two  casks,  one  containing  15  gal.  of  water, 
and  the  other  35  gal.  of  spirits ;  how  many  gallons  must  be 
transferred  from  each  to  the  other  in  order  that  the  mix- 
tures in  each  may  be  of  the  same  strength  ? 

191.  Find    the    value     to     three     decimal    places     of 

V(0.146)2+ (0.063)2. 

3 

192.  Find  the  value  of  — ==: ,  correct  to  four  places  of 

decimals  V19  —  4 

.go    a»    „| -...(2.01  +2.25  X  0.004W(1.0337- 31.09  x 0.03) 
^    ^  4.5-900 


2fof  lH  +  4t 
194.    Simplify^," 


MISCELLANEOUS  EXAMPLES.  335 

195.  Find  the  sum  of  - — ^ — --—,  and  ^  ^  ,J^  ^  ,,^}^  * 

f  X|x(ir  1.6  +  0.625 

196.  Simplify  the  following  expressions :  V2J,  -v/JU, 
^mi,  and  {2iy. 

197.  A  and  B  together  have  f  136,  and  -|  of  A's  money  is 
equal  to  f  of  B's.     How  much  has  each  ? 

198.  A  certain  piece  of  work  can  be  done  by  8  men  or  16 
boys  in  10  days.  In  how  many  days  can  the  work  be  done 
by  8  men  and  16  boys  ? 

199.  If  8  horses  in  30  days  eat  3^  tons  of  hay,  how  long 
will  4y9^  tons  last  10  horses,  15  cows,  and  10  sheep,  each 
cow  eating  |  a&  much  as  a  horse,  and  each  sheep  J  as  much 
as  a  cow  ? 

200.  A  pail  will  hold  5^.  The  area  of  its  base  is  330'^"". 
Kequired  its  height  in  inches. 

201.  One  meter  equals  39.4  inches.  How  many  cubic 
inches  are  there  in  one  liter  ? 

202.  Leap  year  is  omitted  once  in  every  century  except 
in  those  centuries  whose  number  is  divisible  by  four.  What 
is  the  average  length  of  a  year  ? 

203.  What  is  the  value  in  pounds  sterling  of  half  an  acre 
of  land  at  9^  pence  per  square  foot  ? 

204.  Separate  280  into  two  such  numbers  that  -f  of  one  is 
equal  to  the  other. 

205.  A  milkman  bought  40  gal.  of  new  milk  at  16  cents  a 
gallon  and  60  gal.  of  skimmed  milk  at  8  cents  a  gallon, 
which  he  mixed  with  12  gal.  of  water,  and  sold  the  whole  at 
24  cents  a  gallon.     What  was  his  profit  ? 

206.  What  sum  placed  at  simple  interest  for  3  yr.  10  mo. 
at  7%  will  amount  to  the  same  as  $1500  placed  at  compound 
interest  for  the  same  time  at  7|^^  ? 


336  ARITHMETIC. 

207.  I  buy  goods  to  the  amount  of  $4978.70,  payable  in 
4  mo.  with  interest  at  5%,  and  give  my  note  without  in- 
terest.    What  must  be  the  face  of  the  note  ? 

208.  Compute  the  value  of  VS  —  1  +  V6  to  four  decimal 
places. 

209.  Extract  the  square  root  of  2.26  to  three  places  of 
decimals.  Show  how  you  can  derive  from  the  square  root 
of  this  number  that  of  0.0226. 

210.  A  rectangle  is  1.25^"*  long  and  3.5^™  wide  ;  find  the 
side  of  the  equivalent  square  in  dekameters. 

211.  What  is  the  length  of  a  cubical  bin  which  will  con- 
tain 4500  cu.  ft.  ? 

212.  Find  as  circulating  decimals  the  square  of  0.4  and 
the  square  root  of  0.694. 

213.  A  rectangular  block  of  stone,  square  at  the  base  and 
8  ft.  high,  contains  162  cu.  ft.  What  is  the  length  of  one 
side  of  the  base  ? 

214.  The  weight  of  a  cubical  block  of  stone,  2  ft.  on  each 
edge,  is  1352  lb.  What  is  the  weight  of  a  cubical  block 
whose  edge  is  4  ft.  ? 

215.  A  cubical  vessel  contains  150  lb.  of  pure  water. 
Find  the  length  of  an  inner  edge  of  the  vessel  in  decimeters. 

216.  How  many  books,  each  10|-  in.  long,  4^  in.  wide,  and 
If  in.  thick,  can  be  packed  in  a  box  5  ft.  3  in.  long,  3  ft. 
wide,  and  2  ft.  9  in.  thick  V 

217.  Supposing  that  the  driving  wheels  of  a  locomotive 
are  16  ft.  in  circumference,  what  number  of  revolutions 
must  they  make  per  minute  so  that  the  locomotive  may 
attain  a  speed  of  60  mi.  per  hour  ? 

218.  rind  wliat  decimal  part  the  square  root  of  -ffj  is  of 
the  square  root  of  5J. 


MISCELLANEOUS  EXAMPLES.  337 

219.  A  sum  of  £250  17  s.  Gd.  is  transmitted  through 
Paris  to  New  York ;  find  the  value  of  the  sum  in  United 
States  money  (£1=24.79  francs  ;  9.2  francs  =  f  1.75). 

220.  What  sum  of  money  is  the  same  part  of  £14  7  s. 
9f  d.  that  4  oz.  7  pwt.  5  gr.  is  of  8  oz.  10  pwt.  15  gr.  ? 

221.  The  wages  of  A  and  B  together  for  22  days  amount 
to  the  same  sum  as  the  wages  of  A  alone  for  38^  days  ;  for 
how  many  days  will  this  sum  pay  the  wages  of  B  alone  ? 

222.  If  Greenwich  time  be  5  hi*.  8  min.  12  sec.  later  than 
Washington,  what  is  the  difference  in  time  between  Wash- 
ington and  a  point  87°  35'  west  of  Greenwich  ? 

223.  How  much  carpeting  f  yd.  wide  will  cover  the  top 
and  sides  of  a  box  3  ft.  6  in.  long,  2  ft.  3  in.  wide,  and  9  in. 

high  ? 

224.  What  would  it  cost  to  paper  the  walls  of  a  room 
30  ft.  8  in.  long,  20  ft.  4  in.  wide,  and  11  ft.  high,  the  paper 
being  2  ft.  3  in.  wide,  and  costing  63  ots.  per  roll  of  12  yd.  ? 

225.  Two  bells  commence  tolling  together,  one  at  the 
rate  of  5  times  in  24  sec,  the  other  at  the  rate  of  4  times  in 
23  sec. ;  in  what  time  will  they  again  toll  together  ? 

226.  A  man  bought  200™  of  cloth  in  France  at  16^  francs 
per  meter ;  he  paid  12^  cents  a  yard  for  duty  and  freight, 
and  sold  it  in  Boston  at  $4.62^  a  yard.  What  was  the 
gain  ? 

227.  A  block  in  the  form  of  a  perfect  cube  contains 
12516  cu.  in.  How  many  square  yards  of  paper  are  required 
to  cover  it  ? 

22S.  Sold  a  hundred  bushels  of  wheat,  which  cost  $150, 
at  50  cents  a  peck,  taking  in  payment  a  6  months'  note 
which  was  discounted  immediately  at  the  bank  at  6%. 
What  was  the  profit  ? 


338  ARITHMETIC. 

229.  A  note  for  $1000,  with  interest  at  7%  payable  an- 
nually, has  run  3  years,  but  no  interest  has  been  paid. 
What  is  now  the  amount  of  the  note  at  simple  interest  ?  at 
annual  interest  ?  at  compound  interest  ? 

230.  What  is  the  compound  interest  of  $1  for  143  yr., 
allowing  it  to  double  once  in  11  yr.  11  mo.  ? 

231.  A  grocer  makes  a  mixture  of  which  21.5  lb.  contain 
^  lb.  of  rye,  12  lb.  of  wheat,  5  lb.  of  oats,  and  4  lb.  of  barley. 
How  much  of  each  ingredient  will  be  contained  in  100  lb. 
of  the  mixture  ? 

232.  If  the  pay  of  a  man,  a  woman,  and  a  boy  be  in  the 
ratio  3,  2,  1 ;  and  24  men,  20  women,  and  16  boys  receive 
£20  8  s.  a  week,  what  will  27  men,  40  women,  and  15  boys 
receive  in  365  days  ? 

233.  Find  how  many  yards  of  carpet  a  yard  wide  must 
be  bought  to  cover  a  floor  23  ft.  6  in.  by  17  ft.  5  in.,  suppos- 
ing that  the  strips  run  lengthwise,  and  that  the  figure  of 
the  carpet  is  8  ft.  long,  and  is  laid  to  match.  Find  also 
how  much  of  the  carpet  must  be  turned  under  or  cut  off  at 
the  ends  and  sides. 

234.  In  the  Centigrade  and  Fahrenheit  thermometers  the 
freezing  points  are  0°  and  32°  respectively,  and  the  boiling 
points  100°  and  212°  respectively.  When  the  Centigrade 
stands  at  37°,  what  will  the  Fahrenheit  read  ? 

235.  A  train  travels  82  mi.  7  fur.  26  rd.  4  yd.  in  3  hr.  48 
min.  51|-  sec. ;  what  is  the  rate  per  hour  ? 

236.  A  person  by  selling  an  article,  which  cost  him  $60 
per  100  pounds,  at  67|-  cents  per  pound,  makes  5%  more 
than  he  would  by  selling  the  whole  for  f267.67|;  how 
many  pounds  were  there  ? 

.  237.  The  amount  of  a  certain  principal  at  a  certain  rate 
of  interest  for  6  mo.  is  $949.76,  and  for  1  yr.  at  the  same 
rate  is  $1003.52.     Eequired  the  rate  per  cent  and  principal 


MISCELLANEOUS   EXAMPLES.  339 

238.  Lead  is  11.4  times,  and  zinc  7.2  times,  as  heavy  as 
water.  If  3  lb.  of  lead  and  2  lb.  of  zinc  be  melted  together, 
compare  the  weight  of  the  alloy  with  that  of  water. 

239.  There  is  a  rectangular  lot  of  ground  64.8  rd.  long 
and  36.05  rd.  widep  and  a  square  lot  of  the  same  area ; 
which  will  require  the  more  feet  of  fencing,  and  how 
much  ? 

240.  A  cubic  foot  of  iron  weighs  450  lb. ;  what  will  be 
the  weight  of  a  rectangular  closed  box  made  of  iron  ^  of 
an  inch  thick,  the  extreme  dimensions  of  the  box  being 
7  ft.  5  in.,  8  ft.  3  in.,  and  4  ft.  3  in.  ? 

241.  In  100.93*^  of  chemically  pure  saltpetre  there  are 
39.04«  of  potassium,  14.01«  of  nitrogen,  and  47.88«  of  oxy- 
gen ;  determine  the  per  cent  of  each  of  these  elements  in 
the  compound,  and  how  many  grams  of  each  there  are  in  a 
kilogram  of  the  latter. 

242.  A  commission  merchant  sells  28000  lb.  of  cotton  at 
12^  cents  per  pound ;  after  deducting  $35.36  for  freight 
and  cartage,  $10.50  for  storage,  and  his  commission,  he 
remits  $3252.89  as  net  proceeds  of  the  sale.  At  what  rate 
did  he  charge  commission  ? 

243.  I  have  bought  a  farm  for  $6500 ;  $2000  of  this  is 
to  be  paid  down,  $500  in  one  year,  and  the  remainder  in 
two  years.  If  a  note  for  the  whole  amount  were  preferred, 
when  would  it  become  due  ? 

244.  On  the  first  of  January,  1884,  A,  B,  and  C  enter 
into  a  partnership.  A  and  B  each  furnish  $4000,  and  C 
$8000.  At  the  end  of  a  year  B  withdraws  $1500,  while  6 
months  later  C  adds  $2000.  At  the  end  of  2  years  they 
find  their  profits  are  $1580.  How  shall  the  profits  be  di- 
vided between  them  ?  What  per  cent  do  they  realize  on 
their  capital  ? 


340  ARITHMETIC. 

245.  If  the  diameter  of  the  earth  is  7926  mi.,  what  height 
in  inches  on  a  globe  2  ft.  in  diameter  will  represent  a  moun- 
tain 15000  ft.  in  height  ? 

246.  The  least  common  multiple  of  four  numbers  is 
283500.  Prove  that  if  three  of  the  iftimbers  are  140,  42, 
and  60,  the  fourth  must  contain  225  as  a  factor. 

247.  A  house  costs  $5000,  and  rents  for  $25  a  month, 
with  $25  to  pay  annually  for  repairs  and  f  50  for  taxes ; 
what  is  the  difference  in  the  income  from  this  and  from  the 
same  money  invested  in  6%  stock  at  96  ? 

248.  A  cistern  contains  23104^^  of  water.  What  is  its 
volume  in  cubic  meters  ?  If  it  has  a  square  base,  and  its 
depth  is  25™,  what  is  the  length  of  an  edge  of  its  base  ? 

249.  How  many  five-cent  coins  may  be  made  from  a  bar 
of  silver  O.S'"  long,  0.6'^*"  wide,  and  5*=™  thick,  if  each  coin 
weighs  5^,  and  silver  is  10.5  times  as  heavy  as  water  ? 

250.  Eesolve  21600  into  its  prime  factors  ;  and  use  them 
to  find  the  greatest  square  number,  and  also  the  greatest 
cube,  that  will  divide  21600  without  remainder. 

6  29   nf  a/Q4  8  

251.  Divide  "^""215  ^y  >/67419143. 

252.  rind  the  value  to  three  decimal  places  of  the  expres- 
sion   3/3^  X  1|  +  4yL  —  3_^ 

\5i-7|-28^-Fi 

253.  The  length  of  a  rectangular  field  is  |  of  the  breadth, 
and  the  area  is  9  acres.  Find  the  diagonal  in  rods,  feet,  and 
inches. 

254.  If  A  can  row  at  the  rate  of  12|^  miles  per  hour,  and 
B  at  the  rate  of  llf  miles  per  hour,  what  start  should  A 
give  B  in  a  race  of  500  yards  in  order  to  beat  him  by  one 
yard  ? 


MISCELLAKEOtJS  EXAMPLES.  S41 

255.  A  clock  gains  3^  min.  in  23  hr.  59  min.  45  sec.  ;  at 
noon  it  is  2  min.  slow  ;  when  will  it  indicate  correct  time  ? 

256.  5  cu.  ft.  of  gold  weigh  98.2  times  as  much  as  a  cubic 
foot  of  water,  and  2  cu.  ft.  of  copper  weigh  18  times  as 
much  as  a  cubic  foot  of  water ;  how  many  cubic  inches  of 
copper  will  weigh  as  much  as  J  of  a  cubic  inch  of  gold  ? 

257.  A  wins  9  games  out  of  15  when  playing  against  B, 
and  IG  out  of  25  when  playing  against  C.  How  many 
games  out  of  118  should  C  win  when  playing  against  B  ? 

258.  A  cube  is  formed  of  a  certain  number  of  pounds 
Avoirdupois  of  a  substance,  and  the  same  number  of  pounds 
Troy  of  the  same  substance.  What  rjftio  will  a  side  of  the 
(jube  bear  to  a  side  of  a  cube  formed  of  the  same  number  of 
pounds  as  before,  but  all  Avoirdupois?  (1751b.  Troy  =  144 
lb.  Avoirdupois). 

259.  A  Frenchman  sells  a  draft  on  Pai-is  for  10000  francs 
in  New  York  at  5.15  francs  for  f  1,  and  witli  the  proceeds 
buys  a  bill  of  exchange  on  London  at  8^%  premium;  what 
is  the  amount  of  the  bill  in  English  currency  ? 

260.  A  man  paid  |  of  his  money  for  stock,  J  of  what  re- 
mained for  goods,  and  ^  of  what  then  remained  for  tools ; 
he  then  found  that  $26  was  ^  of  one  half  of  what  was  left. 
Find  what  part  of  the  whole  was  left,  and  how  much  money 
he  had  at  first. 

261.  Find  in  acres  the  area  of  a  rectangular  field  of 
which  the  longer  side  is  to  the  shorter  as  15 : 8,  and  which 
a  person  walking  at  the  rate  of  3^^  miles  per  hour  takes 
5  min.  45  sec.  to  walk  around. 

262.  A  rectangular  piece  of  ground  is  13  ch.  44  li.  by  8  ch. 
40  li.  How  many  square  feet  would  be  occupied  on  paper 
by  a  plan  of  the  land  drawn  upon  a  scale  of  IJ  inches  to  a 
chain  ? 


342  ARITHMETIC. 

263.  A  and  B  run  a  race,  their  rates  of  running  being  as 
17  to  18.  A  runs  2 J  mi.  in  16  min.  48  sec.,  and  B  runs  the 
entire  distance  in  34  min.     What  was  the  entire  distance  ? 


264.  Mr.  A.  buys  a  house  for  $10000  and  rents  it  for 
a  month,  paying  $150  per  annum  for  taxes  and  repairs. 
He  also  buys  181  shares  of  railroad  stock  (par  $50)  for  $55 
each,  and  receives  a  dividend  of  7%.  What  is  his  income 
and  rate  of  interest  from  each  investment  ?  What  is  his 
total  investment,  income,  and  rate  of  interest  ? 

265.  Find  to  two  decimal  places  the  sixth  root  of  the 
least  common  multiple  of  899  and  1073. 

266.  A  sum  of  money  was  placed  at  interest  at  6%  per 
annum ;  at  the  end  of  the  first  year  the  interest  was  added 
to  the  principal,  and  at  the  end  of  the  second  year  the 
amount  was  $842.70  ;  what  was  the  original  sum  ? 

267.  Six  men,  working  9  hours  a  day,  can  do  a  piece  of 
work  in  15  days.  In  how  many  days  will  a  party  of  men, 
working  10  hours  a  day,  do  the  work,  the  number  of  men 
being  equal  to  the  number  of  days  ? 

268.  A  square  of  25™  on  a  side  is  inscribed  in  a  circular 
walk  5™  wide.  This  walk  is  covered  with  asphalt  10*""  thick. 
What  is  the  weight  of  the  asphalt  in  metric  tons  if  its  spe- 
cific gravity  is  10  ? 

269.  A  owes  B  $1080  due  July  5th,  1885,  $250  due  Sept. 
15th,  1885,  $700  due  Dec.  10th,  1885,  and  $300  due  Mar. 
20th,  1886;  B  owes  A  $500  due  Aug.  25th,  1885,  $400  due 
Jan.  1st,  1886,  and  $350  due  June  20th,  1886.  Eind  the 
time  when  the  balance  due  B  may  be  paid  without  loss  to 
either  party.  Find  also  the  equitable  value  of  that  balance 
if  payment  were  made  Sept.  15th,  1886,  the  rate  of  interest 
being  6%. 


MISCELLANEOUS  EXAMPLES.  343 

270.  A  loaded  wagon  weighs  2  T.  3  cwt.  48  lb. ;  the  wagon 
itself  weighs  18  cwt.  75  lb.  The  load  consists  of  215  pack- 
ages, each  of  the  same  weight.  Find  the  weight  of  each, 
and  reduce  it  to  kilograms. 

271.  A  box  with-  a  lid  measures  externally  16  in.  each 
way,  and  the  wood  of  which  it  is  made  is  1  in.  thick ;  what 
would  be  the  weight  of  the  box  when  filled  with  paper,  a 
cubic  foot  of  paper  weighing  792  oz.  and  a  cubic  foot  of 
wood  840  oz.  ? 

272.  A  box  in  the  form  of  a  cube  is  partially  filled  with 
water.  Half  a  dozen  balls,  each  4  in.  in  diameter,  are  thrown 
in,  and  the  water  rises  i  in.  in  consequence.  Find  the  length 
of  an  edge  of  the  box. 

273.  English  shillings  are  coined  from  a  metal  which  con- 
tains 37  parts  of  silver  to  3  parts  of  alloy ;  one  pound  of  this 
metal  is  coined  into  66  shillings.  The  United  States  dollar 
weighs  412.5  grains,  and  consists  of  9  parts  silver  to  1  of 
alloy.  What  fraction  of  the  United  States  dollar  will  con- 
tain the  same  amount  of  silver  as  one  English  shilling  ? 

274.  Two  bodies  let  fall  at  different  instants  from  the 
same  point  are  found,  VW^^^  seconds  after  the  latter  of 
them  started,  to  have  fallen,  the  one  25™,  the  other  100™. 
These  distances  being  to  one  another  as  the  squares  of  the 
times  during  which  the  bodies  have  been  falling,  how  many 
seconds  must  the  one  body  have  started  before  the  other  ? 

275.  Using  the  prefix  "mega-^'  for  a  million  times,  and 
"micro-^^  for  a  millionth  part  of.  show  how  many  megame- 
ters  (roughly)  make  up  the  earth's  circumference,  and  how 
many  cubic  micrometers  of  water  weigh  a  microgram. 


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THE  UNIVERSITY  OF  CALIFORNIA  UBRARY 


